3.23.51 \(\int (d+e x)^{3/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\) [2251]

Optimal. Leaf size=424 \[ -\frac {256 (2 c d-b e)^4 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac {128 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac {32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2} \]

[Out]

-256/765765*(-b*e+2*c*d)^4*(-10*b*e*g+3*c*d*g+17*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^6/e^2/(e*x+d)
^(7/2)-128/109395*(-b*e+2*c*d)^3*(-10*b*e*g+3*c*d*g+17*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^5/e^2/(
e*x+d)^(5/2)-32/12155*(-b*e+2*c*d)^2*(-10*b*e*g+3*c*d*g+17*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^4/e
^2/(e*x+d)^(3/2)-2/17*g*(e*x+d)^(3/2)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c/e^2-16/3315*(-b*e+2*c*d)*(-10*b
*e*g+3*c*d*g+17*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^3/e^2/(e*x+d)^(1/2)-2/255*(-10*b*e*g+3*c*d*g+1
7*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)*(e*x+d)^(1/2)/c^2/e^2

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Rubi [A]
time = 0.47, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {808, 670, 662} \begin {gather*} -\frac {256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac {128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac {32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-10 b e g+3 c d g+17 c e f)}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(-256*(2*c*d - b*e)^4*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(765765*c^6
*e^2*(d + e*x)^(7/2)) - (128*(2*c*d - b*e)^3*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(109395*c^5*e^2*(d + e*x)^(5/2)) - (32*(2*c*d - b*e)^2*(17*c*e*f + 3*c*d*g - 10*b*e*g)*(d*(c*d - b
*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(12155*c^4*e^2*(d + e*x)^(3/2)) - (16*(2*c*d - b*e)*(17*c*e*f + 3*c*d*g - 10
*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3315*c^3*e^2*Sqrt[d + e*x]) - (2*(17*c*e*f + 3*c*d*g - 1
0*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(255*c^2*e^2) - (2*g*(d + e*x)^(3/2)*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(17*c*e^2)

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 808

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rubi steps

\begin {align*} \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx &=-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{17 c e^3}\\ &=-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}+\frac {(8 (2 c d-b e) (17 c e f+3 c d g-10 b e g)) \int \sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{255 c^2 e}\\ &=-\frac {16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}+\frac {\left (16 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx}{1105 c^3 e}\\ &=-\frac {32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}+\frac {\left (64 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{12155 c^4 e}\\ &=-\frac {128 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac {32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}+\frac {\left (128 (2 c d-b e)^4 (17 c e f+3 c d g-10 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{109395 c^5 e}\\ &=-\frac {256 (2 c d-b e)^4 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac {128 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac {32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 367, normalized size = 0.87 \begin {gather*} \frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (17 e f+118 d g+35 e g x)-32 b^3 c^2 e^3 \left (2253 d^2 g+7 e^2 x (34 f+45 g x)+2 d e (391 f+756 g x)\right )+16 b^2 c^3 e^2 \left (10864 d^3 g+294 d e^2 x (17 f+21 g x)+21 e^3 x^2 (51 f+55 g x)+3 d^2 e (2397 f+4249 g x)\right )-2 b c^4 e \left (104843 d^4 g+231 e^4 x^3 (68 f+65 g x)+84 d e^3 x^2 (969 f+968 g x)+42 d^2 e^2 x (3842 f+4287 g x)+4 d^3 e (32623 f+50554 g x)\right )+c^5 \left (94134 d^5 g+3003 e^5 x^4 (17 f+15 g x)+462 d e^4 x^3 (578 f+507 g x)+126 d^2 e^3 x^2 (4471 f+3949 g x)+28 d^3 e^2 x (21097 f+19638 g x)+d^4 e (278171 f+329469 g x)\right )\right )}{765765 c^6 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-1280*b^5*e^5*g + 128*b^4*c*e^4*(17*e*f +
118*d*g + 35*e*g*x) - 32*b^3*c^2*e^3*(2253*d^2*g + 7*e^2*x*(34*f + 45*g*x) + 2*d*e*(391*f + 756*g*x)) + 16*b^2
*c^3*e^2*(10864*d^3*g + 294*d*e^2*x*(17*f + 21*g*x) + 21*e^3*x^2*(51*f + 55*g*x) + 3*d^2*e*(2397*f + 4249*g*x)
) - 2*b*c^4*e*(104843*d^4*g + 231*e^4*x^3*(68*f + 65*g*x) + 84*d*e^3*x^2*(969*f + 968*g*x) + 42*d^2*e^2*x*(384
2*f + 4287*g*x) + 4*d^3*e*(32623*f + 50554*g*x)) + c^5*(94134*d^5*g + 3003*e^5*x^4*(17*f + 15*g*x) + 462*d*e^4
*x^3*(578*f + 507*g*x) + 126*d^2*e^3*x^2*(4471*f + 3949*g*x) + 28*d^3*e^2*x*(21097*f + 19638*g*x) + d^4*e*(278
171*f + 329469*g*x))))/(765765*c^6*e^2*Sqrt[d + e*x])

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Maple [A]
time = 0.06, size = 529, normalized size = 1.25

method	result	size
default	\(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{3} \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 f \,d^{4} c^{5} e \right )}{765765 \sqrt {e x +d}\, c^{6} e^{2}}\)	\(529\)
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 f \,d^{4} c^{5} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{765765 c^{6} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\)	\(535\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/765765/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*x+b*e-c*d)^3*(-45045*c^5*e^5*g*x^5+30030*b*c^4*e
^5*g*x^4-234234*c^5*d*e^4*g*x^4-51051*c^5*e^5*f*x^4-18480*b^2*c^3*e^5*g*x^3+162624*b*c^4*d*e^4*g*x^3+31416*b*c
^4*e^5*f*x^3-497574*c^5*d^2*e^3*g*x^3-267036*c^5*d*e^4*f*x^3+10080*b^3*c^2*e^5*g*x^2-98784*b^2*c^3*d*e^4*g*x^2
-17136*b^2*c^3*e^5*f*x^2+360108*b*c^4*d^2*e^3*g*x^2+162792*b*c^4*d*e^4*f*x^2-549864*c^5*d^3*e^2*g*x^2-563346*c
^5*d^2*e^3*f*x^2-4480*b^4*c*e^5*g*x+48384*b^3*c^2*d*e^4*g*x+7616*b^3*c^2*e^5*f*x-203952*b^2*c^3*d^2*e^3*g*x-79
968*b^2*c^3*d*e^4*f*x+404432*b*c^4*d^3*e^2*g*x+322728*b*c^4*d^2*e^3*f*x-329469*c^5*d^4*e*g*x-590716*c^5*d^3*e^
2*f*x+1280*b^5*e^5*g-15104*b^4*c*d*e^4*g-2176*b^4*c*e^5*f+72096*b^3*c^2*d^2*e^3*g+25024*b^3*c^2*d*e^4*f-173824
*b^2*c^3*d^3*e^2*g-115056*b^2*c^3*d^2*e^3*f+209686*b*c^4*d^4*e*g+260984*b*c^4*d^3*e^2*f-94134*c^5*d^5*g-278171
*c^5*d^4*e*f)/c^6/e^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (396) = 792\).
time = 0.37, size = 1041, normalized size = 2.46 \begin {gather*} \frac {2 \, {\left (3003 \, c^{7} x^{7} e^{7} - 16363 \, c^{7} d^{7} + 64441 \, b c^{6} d^{6} e - 101913 \, b^{2} c^{5} d^{5} e^{2} + 84195 \, b^{3} c^{4} d^{4} e^{3} - 40200 \, b^{4} c^{3} d^{3} e^{4} + 11568 \, b^{5} c^{2} d^{2} e^{5} - 1856 \, b^{6} c d e^{6} + 128 \, b^{7} e^{7} + 231 \, {\left (29 \, c^{7} d e^{6} + 31 \, b c^{6} e^{7}\right )} x^{6} - 63 \, {\left (79 \, c^{7} d^{2} e^{5} - 398 \, b c^{6} d e^{6} - 71 \, b^{2} c^{5} e^{7}\right )} x^{5} - 35 \, {\left (587 \, c^{7} d^{3} e^{4} - 525 \, b c^{6} d^{2} e^{5} - 633 \, b^{2} c^{5} d e^{6} - b^{3} c^{4} e^{7}\right )} x^{4} - 5 \, {\left (835 \, c^{7} d^{4} e^{3} + 6548 \, b c^{6} d^{3} e^{4} - 8586 \, b^{2} c^{5} d^{2} e^{5} - 92 \, b^{3} c^{4} d e^{6} + 8 \, b^{4} c^{3} e^{7}\right )} x^{3} + 3 \, {\left (7339 \, c^{7} d^{5} e^{2} - 20435 \, b c^{6} d^{4} e^{3} + 12250 \, b^{2} c^{5} d^{3} e^{4} + 1030 \, b^{3} c^{4} d^{2} e^{5} - 200 \, b^{4} c^{3} d e^{6} + 16 \, b^{5} c^{2} e^{7}\right )} x^{2} + {\left (14341 \, c^{7} d^{6} e - 21006 \, b c^{6} d^{5} e^{2} - 4395 \, b^{2} c^{5} d^{4} e^{3} + 15180 \, b^{3} c^{4} d^{3} e^{4} - 4920 \, b^{4} c^{3} d^{2} e^{5} + 864 \, b^{5} c^{2} d e^{6} - 64 \, b^{6} c e^{7}\right )} x\right )} \sqrt {-c x e + c d - b e} {\left (x e + d\right )} f}{45045 \, {\left (c^{5} x e^{2} + c^{5} d e\right )}} + \frac {2 \, {\left (45045 \, c^{8} x^{8} e^{8} - 94134 \, c^{8} d^{8} + 492088 \, b c^{7} d^{7} e - 1085284 \, b^{2} c^{6} d^{6} e^{2} + 1316760 \, b^{3} c^{5} d^{5} e^{3} - 962550 \, b^{4} c^{4} d^{4} e^{4} + 436704 \, b^{5} c^{3} d^{3} e^{5} - 121248 \, b^{6} c^{2} d^{2} e^{6} + 18944 \, b^{7} c d e^{7} - 1280 \, b^{8} e^{8} + 3003 \, {\left (33 \, c^{8} d e^{7} + 35 \, b c^{7} e^{8}\right )} x^{7} - 231 \, {\left (303 \, c^{8} d^{2} e^{6} - 1558 \, b c^{7} d e^{7} - 275 \, b^{2} c^{6} e^{8}\right )} x^{6} - 63 \, {\left (4527 \, c^{8} d^{3} e^{5} - 4129 \, b c^{7} d^{2} e^{6} - 4813 \, b^{2} c^{6} d e^{7} - 5 \, b^{3} c^{5} e^{8}\right )} x^{5} - 35 \, {\left (1761 \, c^{8} d^{4} e^{4} + 11860 \, b c^{7} d^{3} e^{5} - 15954 \, b^{2} c^{6} d^{2} e^{6} - 108 \, b^{3} c^{5} d e^{7} + 10 \, b^{4} c^{4} e^{8}\right )} x^{4} + 5 \, {\left (51549 \, c^{8} d^{5} e^{3} - 146429 \, b c^{7} d^{4} e^{4} + 91238 \, b^{2} c^{6} d^{3} e^{5} + 4506 \, b^{3} c^{5} d^{2} e^{6} - 944 \, b^{4} c^{4} d e^{7} + 80 \, b^{5} c^{3} e^{8}\right )} x^{3} + 3 \, {\left (52047 \, c^{8} d^{6} e^{2} - 89650 \, b c^{7} d^{5} e^{3} + 15875 \, b^{2} c^{6} d^{4} e^{4} + 30740 \, b^{3} c^{5} d^{3} e^{5} - 10900 \, b^{4} c^{4} d^{2} e^{6} + 2048 \, b^{5} c^{3} d e^{7} - 160 \, b^{6} c^{2} e^{8}\right )} x^{2} - {\left (47067 \, c^{8} d^{7} e - 198977 \, b c^{7} d^{6} e^{2} + 343665 \, b^{2} c^{6} d^{5} e^{3} - 314715 \, b^{3} c^{5} d^{4} e^{4} + 166560 \, b^{4} c^{4} d^{3} e^{5} - 51792 \, b^{5} c^{3} d^{2} e^{6} + 8832 \, b^{6} c^{2} d e^{7} - 640 \, b^{7} c e^{8}\right )} x\right )} \sqrt {-c x e + c d - b e} {\left (x e + d\right )} g}{765765 \, {\left (c^{6} x e^{3} + c^{6} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*c^7*x^7*e^7 - 16363*c^7*d^7 + 64441*b*c^6*d^6*e - 101913*b^2*c^5*d^5*e^2 + 84195*b^3*c^4*d^4*e^3
- 40200*b^4*c^3*d^3*e^4 + 11568*b^5*c^2*d^2*e^5 - 1856*b^6*c*d*e^6 + 128*b^7*e^7 + 231*(29*c^7*d*e^6 + 31*b*c
^6*e^7)*x^6 - 63*(79*c^7*d^2*e^5 - 398*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 35*(587*c^7*d^3*e^4 - 525*b*c^6*d^2
*e^5 - 633*b^2*c^5*d*e^6 - b^3*c^4*e^7)*x^4 - 5*(835*c^7*d^4*e^3 + 6548*b*c^6*d^3*e^4 - 8586*b^2*c^5*d^2*e^5 -
92*b^3*c^4*d*e^6 + 8*b^4*c^3*e^7)*x^3 + 3*(7339*c^7*d^5*e^2 - 20435*b*c^6*d^4*e^3 + 12250*b^2*c^5*d^3*e^4 + 1
030*b^3*c^4*d^2*e^5 - 200*b^4*c^3*d*e^6 + 16*b^5*c^2*e^7)*x^2 + (14341*c^7*d^6*e - 21006*b*c^6*d^5*e^2 - 4395*
b^2*c^5*d^4*e^3 + 15180*b^3*c^4*d^3*e^4 - 4920*b^4*c^3*d^2*e^5 + 864*b^5*c^2*d*e^6 - 64*b^6*c*e^7)*x)*sqrt(-c*
x*e + c*d - b*e)*(x*e + d)*f/(c^5*x*e^2 + c^5*d*e) + 2/765765*(45045*c^8*x^8*e^8 - 94134*c^8*d^8 + 492088*b*c^
7*d^7*e - 1085284*b^2*c^6*d^6*e^2 + 1316760*b^3*c^5*d^5*e^3 - 962550*b^4*c^4*d^4*e^4 + 436704*b^5*c^3*d^3*e^5
- 121248*b^6*c^2*d^2*e^6 + 18944*b^7*c*d*e^7 - 1280*b^8*e^8 + 3003*(33*c^8*d*e^7 + 35*b*c^7*e^8)*x^7 - 231*(30
3*c^8*d^2*e^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*x^6 - 63*(4527*c^8*d^3*e^5 - 4129*b*c^7*d^2*e^6 - 4813*b^2
*c^6*d*e^7 - 5*b^3*c^5*e^8)*x^5 - 35*(1761*c^8*d^4*e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3
*c^5*d*e^7 + 10*b^4*c^4*e^8)*x^4 + 5*(51549*c^8*d^5*e^3 - 146429*b*c^7*d^4*e^4 + 91238*b^2*c^6*d^3*e^5 + 4506*
b^3*c^5*d^2*e^6 - 944*b^4*c^4*d*e^7 + 80*b^5*c^3*e^8)*x^3 + 3*(52047*c^8*d^6*e^2 - 89650*b*c^7*d^5*e^3 + 15875
*b^2*c^6*d^4*e^4 + 30740*b^3*c^5*d^3*e^5 - 10900*b^4*c^4*d^2*e^6 + 2048*b^5*c^3*d*e^7 - 160*b^6*c^2*e^8)*x^2 -
(47067*c^8*d^7*e - 198977*b*c^7*d^6*e^2 + 343665*b^2*c^6*d^5*e^3 - 314715*b^3*c^5*d^4*e^4 + 166560*b^4*c^4*d^
3*e^5 - 51792*b^5*c^3*d^2*e^6 + 8832*b^6*c^2*d*e^7 - 640*b^7*c*e^8)*x)*sqrt(-c*x*e + c*d - b*e)*(x*e + d)*g/(c
^6*x*e^3 + c^6*d*e^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (396) = 792\).
time = 2.84, size = 1067, normalized size = 2.52 \begin {gather*} -\frac {2 \, {\left (94134 \, c^{8} d^{8} g - {\left (45045 \, c^{8} g x^{8} + 2176 \, b^{7} c f - 1280 \, b^{8} g + 3003 \, {\left (17 \, c^{8} f + 35 \, b c^{7} g\right )} x^{7} + 231 \, {\left (527 \, b c^{7} f + 275 \, b^{2} c^{6} g\right )} x^{6} + 63 \, {\left (1207 \, b^{2} c^{6} f + 5 \, b^{3} c^{5} g\right )} x^{5} + 35 \, {\left (17 \, b^{3} c^{5} f - 10 \, b^{4} c^{4} g\right )} x^{4} - 40 \, {\left (17 \, b^{4} c^{4} f - 10 \, b^{5} c^{3} g\right )} x^{3} + 48 \, {\left (17 \, b^{5} c^{3} f - 10 \, b^{6} c^{2} g\right )} x^{2} - 64 \, {\left (17 \, b^{6} c^{2} f - 10 \, b^{7} c g\right )} x\right )} e^{8} - {\left (99099 \, c^{8} d g x^{7} - 31552 \, b^{6} c^{2} d f + 18944 \, b^{7} c d g + 231 \, {\left (493 \, c^{8} d f + 1558 \, b c^{7} d g\right )} x^{6} + 63 \, {\left (6766 \, b c^{7} d f + 4813 \, b^{2} c^{6} d g\right )} x^{5} + 105 \, {\left (3587 \, b^{2} c^{6} d f + 36 \, b^{3} c^{5} d g\right )} x^{4} + 20 \, {\left (391 \, b^{3} c^{5} d f - 236 \, b^{4} c^{4} d g\right )} x^{3} - 24 \, {\left (425 \, b^{4} c^{4} d f - 256 \, b^{5} c^{3} d g\right )} x^{2} + 96 \, {\left (153 \, b^{5} c^{3} d f - 92 \, b^{6} c^{2} d g\right )} x\right )} e^{7} + 3 \, {\left (23331 \, c^{8} d^{2} g x^{6} - 65552 \, b^{5} c^{3} d^{2} f + 40416 \, b^{6} c^{2} d^{2} g + 21 \, {\left (1343 \, c^{8} d^{2} f - 4129 \, b c^{7} d^{2} g\right )} x^{5} - 35 \, {\left (2975 \, b c^{7} d^{2} f + 5318 \, b^{2} c^{6} d^{2} g\right )} x^{4} - 10 \, {\left (24327 \, b^{2} c^{6} d^{2} f + 751 \, b^{3} c^{5} d^{2} g\right )} x^{3} - 10 \, {\left (1751 \, b^{3} c^{5} d^{2} f - 1090 \, b^{4} c^{4} d^{2} g\right )} x^{2} + 8 \, {\left (3485 \, b^{4} c^{4} d^{2} f - 2158 \, b^{5} c^{3} d^{2} g\right )} x\right )} e^{6} + {\left (285201 \, c^{8} d^{3} g x^{5} + 683400 \, b^{4} c^{4} d^{3} f - 436704 \, b^{5} c^{3} d^{3} g + 35 \, {\left (9979 \, c^{8} d^{3} f + 11860 \, b c^{7} d^{3} g\right )} x^{4} + 10 \, {\left (55658 \, b c^{7} d^{3} f - 45619 \, b^{2} c^{6} d^{3} g\right )} x^{3} - 30 \, {\left (20825 \, b^{2} c^{6} d^{3} f + 3074 \, b^{3} c^{5} d^{3} g\right )} x^{2} - 60 \, {\left (4301 \, b^{3} c^{5} d^{3} f - 2776 \, b^{4} c^{4} d^{3} g\right )} x\right )} e^{5} + 5 \, {\left (12327 \, c^{8} d^{4} g x^{4} - 286263 \, b^{3} c^{5} d^{4} f + 192510 \, b^{4} c^{4} d^{4} g + {\left (14195 \, c^{8} d^{4} f + 146429 \, b c^{7} d^{4} g\right )} x^{3} + 3 \, {\left (69479 \, b c^{7} d^{4} f - 3175 \, b^{2} c^{6} d^{4} g\right )} x^{2} + 3 \, {\left (4981 \, b^{2} c^{6} d^{4} f - 20981 \, b^{3} c^{5} d^{4} g\right )} x\right )} e^{4} - 3 \, {\left (85915 \, c^{8} d^{5} g x^{3} - 577507 \, b^{2} c^{6} d^{5} f + 438920 \, b^{3} c^{5} d^{5} g + {\left (124763 \, c^{8} d^{5} f - 89650 \, b c^{7} d^{5} g\right )} x^{2} - 3 \, {\left (39678 \, b c^{7} d^{5} f + 38185 \, b^{2} c^{6} d^{5} g\right )} x\right )} e^{3} - {\left (156141 \, c^{8} d^{6} g x^{2} + 1095497 \, b c^{7} d^{6} f - 1085284 \, b^{2} c^{6} d^{6} g + {\left (243797 \, c^{8} d^{6} f + 198977 \, b c^{7} d^{6} g\right )} x\right )} e^{2} + {\left (47067 \, c^{8} d^{7} g x + 278171 \, c^{8} d^{7} f - 492088 \, b c^{7} d^{7} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{765765 \, {\left (c^{6} x e^{3} + c^{6} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/765765*(94134*c^8*d^8*g - (45045*c^8*g*x^8 + 2176*b^7*c*f - 1280*b^8*g + 3003*(17*c^8*f + 35*b*c^7*g)*x^7 +
231*(527*b*c^7*f + 275*b^2*c^6*g)*x^6 + 63*(1207*b^2*c^6*f + 5*b^3*c^5*g)*x^5 + 35*(17*b^3*c^5*f - 10*b^4*c^4
*g)*x^4 - 40*(17*b^4*c^4*f - 10*b^5*c^3*g)*x^3 + 48*(17*b^5*c^3*f - 10*b^6*c^2*g)*x^2 - 64*(17*b^6*c^2*f - 10*
b^7*c*g)*x)*e^8 - (99099*c^8*d*g*x^7 - 31552*b^6*c^2*d*f + 18944*b^7*c*d*g + 231*(493*c^8*d*f + 1558*b*c^7*d*g
)*x^6 + 63*(6766*b*c^7*d*f + 4813*b^2*c^6*d*g)*x^5 + 105*(3587*b^2*c^6*d*f + 36*b^3*c^5*d*g)*x^4 + 20*(391*b^3
*c^5*d*f - 236*b^4*c^4*d*g)*x^3 - 24*(425*b^4*c^4*d*f - 256*b^5*c^3*d*g)*x^2 + 96*(153*b^5*c^3*d*f - 92*b^6*c^
2*d*g)*x)*e^7 + 3*(23331*c^8*d^2*g*x^6 - 65552*b^5*c^3*d^2*f + 40416*b^6*c^2*d^2*g + 21*(1343*c^8*d^2*f - 4129
*b*c^7*d^2*g)*x^5 - 35*(2975*b*c^7*d^2*f + 5318*b^2*c^6*d^2*g)*x^4 - 10*(24327*b^2*c^6*d^2*f + 751*b^3*c^5*d^2
*g)*x^3 - 10*(1751*b^3*c^5*d^2*f - 1090*b^4*c^4*d^2*g)*x^2 + 8*(3485*b^4*c^4*d^2*f - 2158*b^5*c^3*d^2*g)*x)*e^
6 + (285201*c^8*d^3*g*x^5 + 683400*b^4*c^4*d^3*f - 436704*b^5*c^3*d^3*g + 35*(9979*c^8*d^3*f + 11860*b*c^7*d^3
*g)*x^4 + 10*(55658*b*c^7*d^3*f - 45619*b^2*c^6*d^3*g)*x^3 - 30*(20825*b^2*c^6*d^3*f + 3074*b^3*c^5*d^3*g)*x^2
- 60*(4301*b^3*c^5*d^3*f - 2776*b^4*c^4*d^3*g)*x)*e^5 + 5*(12327*c^8*d^4*g*x^4 - 286263*b^3*c^5*d^4*f + 19251
0*b^4*c^4*d^4*g + (14195*c^8*d^4*f + 146429*b*c^7*d^4*g)*x^3 + 3*(69479*b*c^7*d^4*f - 3175*b^2*c^6*d^4*g)*x^2
+ 3*(4981*b^2*c^6*d^4*f - 20981*b^3*c^5*d^4*g)*x)*e^4 - 3*(85915*c^8*d^5*g*x^3 - 577507*b^2*c^6*d^5*f + 438920
*b^3*c^5*d^5*g + (124763*c^8*d^5*f - 89650*b*c^7*d^5*g)*x^2 - 3*(39678*b*c^7*d^5*f + 38185*b^2*c^6*d^5*g)*x)*e
^3 - (156141*c^8*d^6*g*x^2 + 1095497*b*c^7*d^6*f - 1085284*b^2*c^6*d^6*g + (243797*c^8*d^6*f + 198977*b*c^7*d^
6*g)*x)*e^2 + (47067*c^8*d^7*g*x + 278171*c^8*d^7*f - 492088*b*c^7*d^7*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x
)*e^2)*sqrt(x*e + d)/(c^6*x*e^3 + c^6*d*e^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)**(3/2)*(f + g*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22660 vs. \(2 (396) = 792\).
time = 2.10, size = 22660, normalized size = 53.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

-2/765765*(51051*c^2*d^6*g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b
^2*e^2)/c^2 + (5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((x*e +
d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e))/c^2)*e^(-1) + 255255*c^2*d^6*f*((-(x*e + d)*c + 2*c*d
- b*e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) - 510510*b*c*d^5*f*((-(x*e + d)*c + 2*c
*d - b*e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c)*e - 14586*c^2*d^5*g*((22*sqrt(2*c*d -
b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*
e^3)*e^(-2)/c^3 - (35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c*d
*e - 42*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c*d + 35*(-(x*e + d)*c + 2*c*d - b*e)^(
3/2)*b^2*e^2 + 42*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*e - 15*((x*e + d)*c - 2*c*d
+ b*e)^3*sqrt(-(x*e + d)*c + 2*c*d - b*e))*e^(-2)/c^3)*e + 102102*c^2*d^5*f*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3
*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*
(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e))/c^2
) - 102102*b*c*d^5*g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2
)/c^2 + (5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((x*e + d)*c
- 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e))/c^2) + 255255*b^2*d^4*f*((-(x*e + d)*c + 2*c*d - b*e)^(3/2)
/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c)*e^2 + 7293*c^2*d^4*f*((22*sqrt(2*c*d - b*e)*c^3*d^3
- 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)*e^(-2)/c^
3 - (35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e - 42*((x*e
+ d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c*d + 35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 +
42*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*e - 15*((x*e + d)*c - 2*c*d + b*e)^3*sqrt
(-(x*e + d)*c + 2*c*d - b*e))*e^(-2)/c^3)*e^2 + 43758*b*c*d^4*g*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d
- b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)*e^(-2)/c^3 - (35*(-(x*e
+ d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e - 42*((x*e + d)*c - 2*c*d
+ b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c*d + 35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 + 42*((x*e + d)*
c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*e - 15*((x*e + d)*c - 2*c*d + b*e)^3*sqrt(-(x*e + d)*c +
2*c*d - b*e))*e^(-2)/c^3)*e^2 - 2431*c^2*d^4*g*((26*sqrt(2*c*d - b*e)*c^4*d^4 + 47*sqrt(2*c*d - b*e)*b*c^3*d^
3*e - 78*sqrt(2*c*d - b*e)*b^2*c^2*d^2*e^2 + 56*sqrt(2*c*d - b*e)*b^3*c*d*e^3 - 16*sqrt(2*c*d - b*e)*b^4*e^4)*
e^(-3)/c^4 + (105*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^3*d^3 - 315*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c^2*d^
2*e - 189*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^2*d^2 + 315*(-(x*e + d)*c + 2*c*d -
b*e)^(3/2)*b^2*c*d*e^2 + 378*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*c*d*e - 135*((x
*e + d)*c - 2*c*d + b*e)^3*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c*d - 105*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b^3*e
^3 - 189*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b^2*e^2 + 135*((x*e + d)*c - 2*c*d + b
*e)^3*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*e - 35*((x*e + d)*c - 2*c*d + b*e)^4*sqrt(-(x*e + d)*c + 2*c*d - b*e)
)*e^(-3)/c^4)*e^2 - 306306*b*c*d^4*f*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*
d - b*e)*b^2*e^2)/c^2 + (5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*e -
3*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*c*d - b*e))/c^2)*e + 51051*b^2*d^4*g*((2*sqrt(2*c*d - b
*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(x*e + d)*c + 2*c*d - b*e)
^(3/2)*c*d - 5*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c + 2*
c*d - b*e))/c^2)*e + 29172*b*c*d^3*f*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sq
rt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)*e^(-2)/c^3 - (35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)
*c^2*d^2 - 70*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e - 42*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x*e + d)*c
+ 2*c*d - b*e)*c*d + 35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 + 42*((x*e + d)*c - 2*c*d + b*e)^2*sqrt(-(x
*e + d)*c + 2*c*d - b*e)*b*e - 15*((x*e + d)*c - 2*c*d + b*e)^3*sqrt(-(x*e + d)*c + 2*c*d - b*e))*e^(-2)/c^3)*
e^3 - 29172*b^2*d^3*g*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)
*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)*e^(-2)/c^3 - (35*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(
-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e - 42*...

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Mupad [B]
time = 4.64, size = 1023, normalized size = 2.41 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^3\,x^6\,\sqrt {d+e\,x}\,\left (275\,g\,b^2\,e^2+1558\,g\,b\,c\,d\,e+527\,f\,b\,c\,e^2-303\,g\,c^2\,d^2+493\,f\,c^2\,d\,e\right )}{3315}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}\,\left (-1280\,g\,b^5\,e^5+15104\,g\,b^4\,c\,d\,e^4+2176\,f\,b^4\,c\,e^5-72096\,g\,b^3\,c^2\,d^2\,e^3-25024\,f\,b^3\,c^2\,d\,e^4+173824\,g\,b^2\,c^3\,d^3\,e^2+115056\,f\,b^2\,c^3\,d^2\,e^3-209686\,g\,b\,c^4\,d^4\,e-260984\,f\,b\,c^4\,d^3\,e^2+94134\,g\,c^5\,d^5+278171\,f\,c^5\,d^4\,e\right )}{765765\,c^6\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (-700\,g\,b^4\,c^4\,e^8+7560\,g\,b^3\,c^5\,d\,e^7+1190\,f\,b^3\,c^5\,e^8+1116780\,g\,b^2\,c^6\,d^2\,e^6+753270\,f\,b^2\,c^6\,d\,e^7-830200\,g\,b\,c^7\,d^3\,e^5+624750\,f\,b\,c^7\,d^2\,e^6-123270\,g\,c^8\,d^4\,e^4-698530\,f\,c^8\,d^3\,e^5\right )}{765765\,c^6\,e^3}+\frac {2\,c^2\,e^5\,g\,x^8\,\sqrt {d+e\,x}}{17}+\frac {x^5\,\sqrt {d+e\,x}\,\left (630\,g\,b^3\,c^5\,e^8+606438\,g\,b^2\,c^6\,d\,e^7+152082\,f\,b^2\,c^6\,e^8+520254\,g\,b\,c^7\,d^2\,e^6+852516\,f\,b\,c^7\,d\,e^7-570402\,g\,c^8\,d^3\,e^5-169218\,f\,c^8\,d^2\,e^6\right )}{765765\,c^6\,e^3}+\frac {2\,c\,e^4\,x^7\,\sqrt {d+e\,x}\,\left (35\,b\,e\,g+33\,c\,d\,g+17\,c\,e\,f\right )}{255}+\frac {x^3\,\sqrt {d+e\,x}\,\left (800\,g\,b^5\,c^3\,e^8-9440\,g\,b^4\,c^4\,d\,e^7-1360\,f\,b^4\,c^4\,e^8+45060\,g\,b^3\,c^5\,d^2\,e^6+15640\,f\,b^3\,c^5\,d\,e^7+912380\,g\,b^2\,c^6\,d^3\,e^5+1459620\,f\,b^2\,c^6\,d^2\,e^6-1464290\,g\,b\,c^7\,d^4\,e^4-1113160\,f\,b\,c^7\,d^3\,e^5+515490\,g\,c^8\,d^5\,e^3-141950\,f\,c^8\,d^4\,e^4\right )}{765765\,c^6\,e^3}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (-160\,g\,b^5\,e^5+1888\,g\,b^4\,c\,d\,e^4+272\,f\,b^4\,c\,e^5-9012\,g\,b^3\,c^2\,d^2\,e^3-3128\,f\,b^3\,c^2\,d\,e^4+21728\,g\,b^2\,c^3\,d^3\,e^2+14382\,f\,b^2\,c^3\,d^2\,e^3+37603\,g\,b\,c^4\,d^4\,e+222632\,f\,b\,c^4\,d^3\,e^2-52047\,g\,c^5\,d^5-124763\,f\,c^5\,d^4\,e\right )}{255255\,c^4\,e}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (640\,g\,b^5\,e^5-7552\,g\,b^4\,c\,d\,e^4-1088\,f\,b^4\,c\,e^5+36048\,g\,b^3\,c^2\,d^2\,e^3+12512\,f\,b^3\,c^2\,d\,e^4-86912\,g\,b^2\,c^3\,d^3\,e^2-57528\,f\,b^2\,c^3\,d^2\,e^3+104843\,g\,b\,c^4\,d^4\,e+130492\,f\,b\,c^4\,d^3\,e^2-47067\,g\,c^5\,d^5+243797\,f\,c^5\,d^4\,e\right )}{765765\,c^5\,e^2}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2),x)

[Out]

((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^3*x^6*(d + e*x)^(1/2)*(275*b^2*e^2*g - 303*c^2*d^2*g + 527*
b*c*e^2*f + 493*c^2*d*e*f + 1558*b*c*d*e*g))/3315 + (2*(b*e - c*d)^3*(d + e*x)^(1/2)*(94134*c^5*d^5*g - 1280*b
^5*e^5*g + 2176*b^4*c*e^5*f + 278171*c^5*d^4*e*f - 209686*b*c^4*d^4*e*g + 15104*b^4*c*d*e^4*g - 260984*b*c^4*d
^3*e^2*f - 25024*b^3*c^2*d*e^4*f + 115056*b^2*c^3*d^2*e^3*f + 173824*b^2*c^3*d^3*e^2*g - 72096*b^3*c^2*d^2*e^3
*g))/(765765*c^6*e^3) + (x^4*(d + e*x)^(1/2)*(1190*b^3*c^5*e^8*f - 700*b^4*c^4*e^8*g - 698530*c^8*d^3*e^5*f -
123270*c^8*d^4*e^4*g + 624750*b*c^7*d^2*e^6*f + 753270*b^2*c^6*d*e^7*f - 830200*b*c^7*d^3*e^5*g + 7560*b^3*c^5
*d*e^7*g + 1116780*b^2*c^6*d^2*e^6*g))/(765765*c^6*e^3) + (2*c^2*e^5*g*x^8*(d + e*x)^(1/2))/17 + (x^5*(d + e*x
)^(1/2)*(152082*b^2*c^6*e^8*f + 630*b^3*c^5*e^8*g - 169218*c^8*d^2*e^6*f - 570402*c^8*d^3*e^5*g + 852516*b*c^7
*d*e^7*f + 520254*b*c^7*d^2*e^6*g + 606438*b^2*c^6*d*e^7*g))/(765765*c^6*e^3) + (2*c*e^4*x^7*(d + e*x)^(1/2)*(
35*b*e*g + 33*c*d*g + 17*c*e*f))/255 + (x^3*(d + e*x)^(1/2)*(800*b^5*c^3*e^8*g - 1360*b^4*c^4*e^8*f - 141950*c
^8*d^4*e^4*f + 515490*c^8*d^5*e^3*g - 1113160*b*c^7*d^3*e^5*f + 15640*b^3*c^5*d*e^7*f - 1464290*b*c^7*d^4*e^4*
g - 9440*b^4*c^4*d*e^7*g + 1459620*b^2*c^6*d^2*e^6*f + 912380*b^2*c^6*d^3*e^5*g + 45060*b^3*c^5*d^2*e^6*g))/(7
65765*c^6*e^3) + (2*x^2*(b*e - c*d)*(d + e*x)^(1/2)*(272*b^4*c*e^5*f - 52047*c^5*d^5*g - 160*b^5*e^5*g - 12476
3*c^5*d^4*e*f + 37603*b*c^4*d^4*e*g + 1888*b^4*c*d*e^4*g + 222632*b*c^4*d^3*e^2*f - 3128*b^3*c^2*d*e^4*f + 143
82*b^2*c^3*d^2*e^3*f + 21728*b^2*c^3*d^3*e^2*g - 9012*b^3*c^2*d^2*e^3*g))/(255255*c^4*e) + (2*x*(b*e - c*d)^2*
(d + e*x)^(1/2)*(640*b^5*e^5*g - 47067*c^5*d^5*g - 1088*b^4*c*e^5*f + 243797*c^5*d^4*e*f + 104843*b*c^4*d^4*e*
g - 7552*b^4*c*d*e^4*g + 130492*b*c^4*d^3*e^2*f + 12512*b^3*c^2*d*e^4*f - 57528*b^2*c^3*d^2*e^3*f - 86912*b^2*
c^3*d^3*e^2*g + 36048*b^3*c^2*d^2*e^3*g))/(765765*c^5*e^2)))/(x + d/e)

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