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

Optimal. Leaf size=419 \[ -\frac {256 (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac {128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac {32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{1287 c^4 e^2 \sqrt {d+e x}}-\frac {16 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{429 c^3 e^2}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2} \]

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Rubi [A]  time = 0.73, antiderivative size = 419, 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 = {794, 656, 648} \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 )^{5/2} (-2 b e g+c d g+3 c e f)}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac {128 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac {32 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{1287 c^4 e^2 \sqrt {d+e x}}-\frac {16 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{429 c^3 e^2}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g+c d g+3 c e f)}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-256*(2*c*d - b*e)^4*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(45045*c^6*e^2*
(d + e*x)^(5/2)) - (128*(2*c*d - b*e)^3*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2
))/(9009*c^5*e^2*(d + e*x)^(3/2)) - (32*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(5/2))/(1287*c^4*e^2*Sqrt[d + e*x]) - (16*(2*c*d - b*e)*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(429*c^3*e^2) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(3/2)*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(39*c^2*e^2) - (2*g*(d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2)^(5/2))/(15*c*e^2)

Rule 648

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 656

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 794

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)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{15 c e^3}\\ &=-\frac {2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac {(8 (2 c d-b e) (3 c e f+c d g-2 b e g)) \int (d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{39 c^2 e}\\ &=-\frac {16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac {2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac {\left (16 (2 c d-b e)^2 (3 c e f+c d g-2 b e g)\right ) \int \sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{143 c^3 e}\\ &=-\frac {32 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{1287 c^4 e^2 \sqrt {d+e x}}-\frac {16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac {2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac {\left (64 (2 c d-b e)^3 (3 c e f+c d g-2 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{1287 c^4 e}\\ &=-\frac {128 (2 c d-b e)^3 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac {32 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{1287 c^4 e^2 \sqrt {d+e x}}-\frac {16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac {2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}+\frac {\left (128 (2 c d-b e)^4 (3 c e f+c d g-2 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{9009 c^5 e}\\ &=-\frac {256 (2 c d-b e)^4 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{45045 c^6 e^2 (d+e x)^{5/2}}-\frac {128 (2 c d-b e)^3 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9009 c^5 e^2 (d+e x)^{3/2}}-\frac {32 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{1287 c^4 e^2 \sqrt {d+e x}}-\frac {16 (2 c d-b e) (3 c e f+c d g-2 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2}-\frac {2 (3 c e f+c d g-2 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{39 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 364, normalized size = 0.87 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (-256 b^5 e^5 g+128 b^4 c e^4 (22 d g+3 e f+5 e g x)-32 b^3 c^2 e^3 \left (389 d^2 g+2 d e (63 f+100 g x)+5 e^2 x (6 f+7 g x)\right )+16 b^2 c^3 e^2 \left (1724 d^3 g+3 d^2 e (347 f+515 g x)+30 d e^2 x (19 f+21 g x)+105 e^3 x^2 (f+g x)\right )-2 b c^4 e \left (15191 d^4 g+4 d^3 e (4131 f+5530 g x)+30 d^2 e^2 x (542 f+553 g x)+420 d e^3 x^2 (17 f+16 g x)+105 e^4 x^3 (12 f+11 g x)\right )+c^5 \left (12686 d^5 g+d^4 e (29049 f+31715 g x)+20 d^3 e^2 x (2505 f+2212 g x)+210 d^2 e^3 x^2 (203 f+173 g x)+210 d e^4 x^3 (90 f+77 g x)+231 e^5 x^4 (15 f+13 g x)\right )\right )}{45045 c^6 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-256*b^5*e^5*g + 128*b^4*c*e^4*(3*e*f + 2
2*d*g + 5*e*g*x) - 32*b^3*c^2*e^3*(389*d^2*g + 5*e^2*x*(6*f + 7*g*x) + 2*d*e*(63*f + 100*g*x)) + 16*b^2*c^3*e^
2*(1724*d^3*g + 105*e^3*x^2*(f + g*x) + 30*d*e^2*x*(19*f + 21*g*x) + 3*d^2*e*(347*f + 515*g*x)) - 2*b*c^4*e*(1
5191*d^4*g + 105*e^4*x^3*(12*f + 11*g*x) + 420*d*e^3*x^2*(17*f + 16*g*x) + 30*d^2*e^2*x*(542*f + 553*g*x) + 4*
d^3*e*(4131*f + 5530*g*x)) + c^5*(12686*d^5*g + 231*e^5*x^4*(15*f + 13*g*x) + 210*d*e^4*x^3*(90*f + 77*g*x) +
210*d^2*e^3*x^2*(203*f + 173*g*x) + 20*d^3*e^2*x*(2505*f + 2212*g*x) + d^4*e*(29049*f + 31715*g*x))))/(45045*c
^6*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 8.65, size = 598, normalized size = 1.43 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{5/2} \left (-256 b^5 e^5 g+640 b^4 c e^4 g (d+e x)+2176 b^4 c d e^4 g+384 b^4 c e^5 f-7168 b^3 c^2 d^2 e^3 g-960 b^3 c^2 e^4 f (d+e x)-3072 b^3 c^2 d e^4 f-1120 b^3 c^2 e^3 g (d+e x)^2-4160 b^3 c^2 d e^3 g (d+e x)+11264 b^2 c^3 d^3 e^2 g+9216 b^2 c^3 d^2 e^3 f+9600 b^2 c^3 d^2 e^2 g (d+e x)+1680 b^2 c^3 e^3 f (d+e x)^2+5760 b^2 c^3 d e^3 f (d+e x)+1680 b^2 c^3 e^2 g (d+e x)^3+5040 b^2 c^3 d e^2 g (d+e x)^2-8192 b c^4 d^4 e g-12288 b c^4 d^3 e^2 f-8960 b c^4 d^3 e g (d+e x)-11520 b c^4 d^2 e^2 f (d+e x)-6720 b c^4 d^2 e g (d+e x)^2-2520 b c^4 e^2 f (d+e x)^3-6720 b c^4 d e^2 f (d+e x)^2-2310 b c^4 e g (d+e x)^4-4200 b c^4 d e g (d+e x)^3+2048 c^5 d^5 g+6144 c^5 d^4 e f+2560 c^5 d^4 g (d+e x)+7680 c^5 d^3 e f (d+e x)+2240 c^5 d^3 g (d+e x)^2+6720 c^5 d^2 e f (d+e x)^2+1680 c^5 d^2 g (d+e x)^3+3465 c^5 e f (d+e x)^4+5040 c^5 d e f (d+e x)^3+3003 c^5 g (d+e x)^5+1155 c^5 d g (d+e x)^4\right )}{45045 c^6 e^2 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(5/2)*(6144*c^5*d^4*e*f - 12288*b*c^4*d^3*e^2*f + 9216*b^2*c^3*d
^2*e^3*f - 3072*b^3*c^2*d*e^4*f + 384*b^4*c*e^5*f + 2048*c^5*d^5*g - 8192*b*c^4*d^4*e*g + 11264*b^2*c^3*d^3*e^
2*g - 7168*b^3*c^2*d^2*e^3*g + 2176*b^4*c*d*e^4*g - 256*b^5*e^5*g + 7680*c^5*d^3*e*f*(d + e*x) - 11520*b*c^4*d
^2*e^2*f*(d + e*x) + 5760*b^2*c^3*d*e^3*f*(d + e*x) - 960*b^3*c^2*e^4*f*(d + e*x) + 2560*c^5*d^4*g*(d + e*x) -
8960*b*c^4*d^3*e*g*(d + e*x) + 9600*b^2*c^3*d^2*e^2*g*(d + e*x) - 4160*b^3*c^2*d*e^3*g*(d + e*x) + 640*b^4*c*
e^4*g*(d + e*x) + 6720*c^5*d^2*e*f*(d + e*x)^2 - 6720*b*c^4*d*e^2*f*(d + e*x)^2 + 1680*b^2*c^3*e^3*f*(d + e*x)
^2 + 2240*c^5*d^3*g*(d + e*x)^2 - 6720*b*c^4*d^2*e*g*(d + e*x)^2 + 5040*b^2*c^3*d*e^2*g*(d + e*x)^2 - 1120*b^3
*c^2*e^3*g*(d + e*x)^2 + 5040*c^5*d*e*f*(d + e*x)^3 - 2520*b*c^4*e^2*f*(d + e*x)^3 + 1680*c^5*d^2*g*(d + e*x)^
3 - 4200*b*c^4*d*e*g*(d + e*x)^3 + 1680*b^2*c^3*e^2*g*(d + e*x)^3 + 3465*c^5*e*f*(d + e*x)^4 + 1155*c^5*d*g*(d
+ e*x)^4 - 2310*b*c^4*e*g*(d + e*x)^4 + 3003*c^5*g*(d + e*x)^5))/(45045*c^6*e^2*(d + e*x)^(5/2))

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fricas [B]  time = 0.45, size = 880, normalized size = 2.10 \begin {gather*} -\frac {2 \, {\left (3003 \, c^{7} e^{7} g x^{7} + 231 \, {\left (15 \, c^{7} e^{7} f + 4 \, {\left (11 \, c^{7} d e^{6} + 4 \, b c^{6} e^{7}\right )} g\right )} x^{6} + 63 \, {\left (10 \, {\left (19 \, c^{7} d e^{6} + 7 \, b c^{6} e^{7}\right )} f + {\left (111 \, c^{7} d^{2} e^{5} + 278 \, b c^{6} d e^{6} + b^{2} c^{5} e^{7}\right )} g\right )} x^{5} + 35 \, {\left (3 \, {\left (79 \, c^{7} d^{2} e^{5} + 206 \, b c^{6} d e^{6} + b^{2} c^{5} e^{7}\right )} f - 2 \, {\left (175 \, c^{7} d^{3} e^{4} - 453 \, b c^{6} d^{2} e^{5} - 9 \, b^{2} c^{5} d e^{6} + b^{3} c^{4} e^{7}\right )} g\right )} x^{4} - 5 \, {\left (12 \, {\left (271 \, c^{7} d^{3} e^{4} - 683 \, b c^{6} d^{2} e^{5} - 19 \, b^{2} c^{5} d e^{6} + 2 \, b^{3} c^{4} e^{7}\right )} f + {\left (4087 \, c^{7} d^{4} e^{3} - 4900 \, b c^{6} d^{3} e^{4} - 618 \, b^{2} c^{5} d^{2} e^{5} + 160 \, b^{3} c^{4} d e^{6} - 16 \, b^{4} c^{3} e^{7}\right )} g\right )} x^{3} - 3 \, {\left (3 \, {\left (3169 \, c^{7} d^{4} e^{3} - 3628 \, b c^{6} d^{3} e^{4} - 694 \, b^{2} c^{5} d^{2} e^{5} + 168 \, b^{3} c^{4} d e^{6} - 16 \, b^{4} c^{3} e^{7}\right )} f + 4 \, {\left (542 \, c^{7} d^{5} e^{2} + 11 \, b c^{6} d^{4} e^{3} - 862 \, b^{2} c^{5} d^{3} e^{4} + 389 \, b^{3} c^{4} d^{2} e^{5} - 88 \, b^{4} c^{3} d e^{6} + 8 \, b^{5} c^{2} e^{7}\right )} g\right )} x^{2} + 3 \, {\left (9683 \, c^{7} d^{6} e - 30382 \, b c^{6} d^{5} e^{2} + 37267 \, b^{2} c^{5} d^{4} e^{3} - 23464 \, b^{3} c^{4} d^{3} e^{4} + 8368 \, b^{4} c^{3} d^{2} e^{5} - 1600 \, b^{5} c^{2} d e^{6} + 128 \, b^{6} c e^{7}\right )} f + 2 \, {\left (6343 \, c^{7} d^{7} - 27877 \, b c^{6} d^{6} e + 50517 \, b^{2} c^{5} d^{5} e^{2} - 48999 \, b^{3} c^{4} d^{4} e^{3} + 27648 \, b^{4} c^{3} d^{3} e^{4} - 9168 \, b^{5} c^{2} d^{2} e^{5} + 1664 \, b^{6} c d e^{6} - 128 \, b^{7} e^{7}\right )} g - {\left (6 \, {\left (1333 \, c^{7} d^{5} e^{2} + 1421 \, b c^{6} d^{4} e^{3} - 4142 \, b^{2} c^{5} d^{3} e^{4} + 1724 \, b^{3} c^{4} d^{2} e^{5} - 368 \, b^{4} c^{3} d e^{6} + 32 \, b^{5} c^{2} e^{7}\right )} f - {\left (6343 \, c^{7} d^{6} e - 21534 \, b c^{6} d^{5} e^{2} + 28983 \, b^{2} c^{5} d^{4} e^{3} - 20016 \, b^{3} c^{4} d^{3} e^{4} + 7632 \, b^{4} c^{3} d^{2} e^{5} - 1536 \, b^{5} c^{2} d e^{6} + 128 \, b^{6} c e^{7}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{45045 \, {\left (c^{6} e^{3} x + c^{6} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45045*(3003*c^7*e^7*g*x^7 + 231*(15*c^7*e^7*f + 4*(11*c^7*d*e^6 + 4*b*c^6*e^7)*g)*x^6 + 63*(10*(19*c^7*d*e^
6 + 7*b*c^6*e^7)*f + (111*c^7*d^2*e^5 + 278*b*c^6*d*e^6 + b^2*c^5*e^7)*g)*x^5 + 35*(3*(79*c^7*d^2*e^5 + 206*b*
c^6*d*e^6 + b^2*c^5*e^7)*f - 2*(175*c^7*d^3*e^4 - 453*b*c^6*d^2*e^5 - 9*b^2*c^5*d*e^6 + b^3*c^4*e^7)*g)*x^4 -
5*(12*(271*c^7*d^3*e^4 - 683*b*c^6*d^2*e^5 - 19*b^2*c^5*d*e^6 + 2*b^3*c^4*e^7)*f + (4087*c^7*d^4*e^3 - 4900*b*
c^6*d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*g)*x^3 - 3*(3*(3169*c^7*d^4*e^3 - 3628
*b*c^6*d^3*e^4 - 694*b^2*c^5*d^2*e^5 + 168*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*f + 4*(542*c^7*d^5*e^2 + 11*b*c^6*d
^4*e^3 - 862*b^2*c^5*d^3*e^4 + 389*b^3*c^4*d^2*e^5 - 88*b^4*c^3*d*e^6 + 8*b^5*c^2*e^7)*g)*x^2 + 3*(9683*c^7*d^
6*e - 30382*b*c^6*d^5*e^2 + 37267*b^2*c^5*d^4*e^3 - 23464*b^3*c^4*d^3*e^4 + 8368*b^4*c^3*d^2*e^5 - 1600*b^5*c^
2*d*e^6 + 128*b^6*c*e^7)*f + 2*(6343*c^7*d^7 - 27877*b*c^6*d^6*e + 50517*b^2*c^5*d^5*e^2 - 48999*b^3*c^4*d^4*e
^3 + 27648*b^4*c^3*d^3*e^4 - 9168*b^5*c^2*d^2*e^5 + 1664*b^6*c*d*e^6 - 128*b^7*e^7)*g - (6*(1333*c^7*d^5*e^2 +
1421*b*c^6*d^4*e^3 - 4142*b^2*c^5*d^3*e^4 + 1724*b^3*c^4*d^2*e^5 - 368*b^4*c^3*d*e^6 + 32*b^5*c^2*e^7)*f - (6
343*c^7*d^6*e - 21534*b*c^6*d^5*e^2 + 28983*b^2*c^5*d^4*e^3 - 20016*b^3*c^4*d^3*e^4 + 7632*b^4*c^3*d^2*e^5 - 1
536*b^5*c^2*d*e^6 + 128*b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^6*e^3*x +
c^6*d*e^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.05, size = 535, normalized size = 1.28 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-3003 g \,e^{5} x^{5} c^{5}+2310 b \,c^{4} e^{5} g \,x^{4}-16170 c^{5} d \,e^{4} g \,x^{4}-3465 c^{5} e^{5} f \,x^{4}-1680 b^{2} c^{3} e^{5} g \,x^{3}+13440 b \,c^{4} d \,e^{4} g \,x^{3}+2520 b \,c^{4} e^{5} f \,x^{3}-36330 c^{5} d^{2} e^{3} g \,x^{3}-18900 c^{5} d \,e^{4} f \,x^{3}+1120 b^{3} c^{2} e^{5} g \,x^{2}-10080 b^{2} c^{3} d \,e^{4} g \,x^{2}-1680 b^{2} c^{3} e^{5} f \,x^{2}+33180 b \,c^{4} d^{2} e^{3} g \,x^{2}+14280 b \,c^{4} d \,e^{4} f \,x^{2}-44240 c^{5} d^{3} e^{2} g \,x^{2}-42630 c^{5} d^{2} e^{3} f \,x^{2}-640 b^{4} c \,e^{5} g x +6400 b^{3} c^{2} d \,e^{4} g x +960 b^{3} c^{2} e^{5} f x -24720 b^{2} c^{3} d^{2} e^{3} g x -9120 b^{2} c^{3} d \,e^{4} f x +44240 b \,c^{4} d^{3} e^{2} g x +32520 b \,c^{4} d^{2} e^{3} f x -31715 c^{5} d^{4} e g x -50100 c^{5} d^{3} e^{2} f x +256 b^{5} e^{5} g -2816 b^{4} c d \,e^{4} g -384 b^{4} c \,e^{5} f +12448 b^{3} c^{2} d^{2} e^{3} g +4032 b^{3} c^{2} d \,e^{4} f -27584 b^{2} c^{3} d^{3} e^{2} g -16656 b^{2} c^{3} d^{2} e^{3} f +30382 b \,c^{4} d^{4} e g +33048 b \,c^{4} d^{3} e^{2} f -12686 c^{5} d^{5} g -29049 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 {3}{2}}}{45045 \left (e x +d \right )^{\frac {3}{2}} c^{6} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45045*(c*e*x+b*e-c*d)*(-3003*c^5*e^5*g*x^5+2310*b*c^4*e^5*g*x^4-16170*c^5*d*e^4*g*x^4-3465*c^5*e^5*f*x^4-16
80*b^2*c^3*e^5*g*x^3+13440*b*c^4*d*e^4*g*x^3+2520*b*c^4*e^5*f*x^3-36330*c^5*d^2*e^3*g*x^3-18900*c^5*d*e^4*f*x^
3+1120*b^3*c^2*e^5*g*x^2-10080*b^2*c^3*d*e^4*g*x^2-1680*b^2*c^3*e^5*f*x^2+33180*b*c^4*d^2*e^3*g*x^2+14280*b*c^
4*d*e^4*f*x^2-44240*c^5*d^3*e^2*g*x^2-42630*c^5*d^2*e^3*f*x^2-640*b^4*c*e^5*g*x+6400*b^3*c^2*d*e^4*g*x+960*b^3
*c^2*e^5*f*x-24720*b^2*c^3*d^2*e^3*g*x-9120*b^2*c^3*d*e^4*f*x+44240*b*c^4*d^3*e^2*g*x+32520*b*c^4*d^2*e^3*f*x-
31715*c^5*d^4*e*g*x-50100*c^5*d^3*e^2*f*x+256*b^5*e^5*g-2816*b^4*c*d*e^4*g-384*b^4*c*e^5*f+12448*b^3*c^2*d^2*e
^3*g+4032*b^3*c^2*d*e^4*f-27584*b^2*c^3*d^3*e^2*g-16656*b^2*c^3*d^2*e^3*f+30382*b*c^4*d^4*e*g+33048*b*c^4*d^3*
e^2*f-12686*c^5*d^5*g-29049*c^5*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^6/e^2/(e*x+d)^(3/2)

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maxima [B]  time = 1.10, size = 875, normalized size = 2.09 \begin {gather*} -\frac {2 \, {\left (1155 \, c^{6} e^{6} x^{6} + 9683 \, c^{6} d^{6} - 30382 \, b c^{5} d^{5} e + 37267 \, b^{2} c^{4} d^{4} e^{2} - 23464 \, b^{3} c^{3} d^{3} e^{3} + 8368 \, b^{4} c^{2} d^{2} e^{4} - 1600 \, b^{5} c d e^{5} + 128 \, b^{6} e^{6} + 210 \, {\left (19 \, c^{6} d e^{5} + 7 \, b c^{5} e^{6}\right )} x^{5} + 35 \, {\left (79 \, c^{6} d^{2} e^{4} + 206 \, b c^{5} d e^{5} + b^{2} c^{4} e^{6}\right )} x^{4} - 20 \, {\left (271 \, c^{6} d^{3} e^{3} - 683 \, b c^{5} d^{2} e^{4} - 19 \, b^{2} c^{4} d e^{5} + 2 \, b^{3} c^{3} e^{6}\right )} x^{3} - 3 \, {\left (3169 \, c^{6} d^{4} e^{2} - 3628 \, b c^{5} d^{3} e^{3} - 694 \, b^{2} c^{4} d^{2} e^{4} + 168 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} x^{2} - 2 \, {\left (1333 \, c^{6} d^{5} e + 1421 \, b c^{5} d^{4} e^{2} - 4142 \, b^{2} c^{4} d^{3} e^{3} + 1724 \, b^{3} c^{3} d^{2} e^{4} - 368 \, b^{4} c^{2} d e^{5} + 32 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{15015 \, {\left (c^{5} e^{2} x + c^{5} d e\right )}} - \frac {2 \, {\left (3003 \, c^{7} e^{7} x^{7} + 12686 \, c^{7} d^{7} - 55754 \, b c^{6} d^{6} e + 101034 \, b^{2} c^{5} d^{5} e^{2} - 97998 \, b^{3} c^{4} d^{4} e^{3} + 55296 \, b^{4} c^{3} d^{3} e^{4} - 18336 \, b^{5} c^{2} d^{2} e^{5} + 3328 \, b^{6} c d e^{6} - 256 \, b^{7} e^{7} + 924 \, {\left (11 \, c^{7} d e^{6} + 4 \, b c^{6} e^{7}\right )} x^{6} + 63 \, {\left (111 \, c^{7} d^{2} e^{5} + 278 \, b c^{6} d e^{6} + b^{2} c^{5} e^{7}\right )} x^{5} - 70 \, {\left (175 \, c^{7} d^{3} e^{4} - 453 \, b c^{6} d^{2} e^{5} - 9 \, b^{2} c^{5} d e^{6} + b^{3} c^{4} e^{7}\right )} x^{4} - 5 \, {\left (4087 \, c^{7} d^{4} e^{3} - 4900 \, b c^{6} d^{3} e^{4} - 618 \, b^{2} c^{5} d^{2} e^{5} + 160 \, b^{3} c^{4} d e^{6} - 16 \, b^{4} c^{3} e^{7}\right )} x^{3} - 12 \, {\left (542 \, c^{7} d^{5} e^{2} + 11 \, b c^{6} d^{4} e^{3} - 862 \, b^{2} c^{5} d^{3} e^{4} + 389 \, b^{3} c^{4} d^{2} e^{5} - 88 \, b^{4} c^{3} d e^{6} + 8 \, b^{5} c^{2} e^{7}\right )} x^{2} + {\left (6343 \, c^{7} d^{6} e - 21534 \, b c^{6} d^{5} e^{2} + 28983 \, b^{2} c^{5} d^{4} e^{3} - 20016 \, b^{3} c^{4} d^{3} e^{4} + 7632 \, b^{4} c^{3} d^{2} e^{5} - 1536 \, b^{5} c^{2} d e^{6} + 128 \, b^{6} c e^{7}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{45045 \, {\left (c^{6} e^{3} x + c^{6} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15015*(1155*c^6*e^6*x^6 + 9683*c^6*d^6 - 30382*b*c^5*d^5*e + 37267*b^2*c^4*d^4*e^2 - 23464*b^3*c^3*d^3*e^3
+ 8368*b^4*c^2*d^2*e^4 - 1600*b^5*c*d*e^5 + 128*b^6*e^6 + 210*(19*c^6*d*e^5 + 7*b*c^5*e^6)*x^5 + 35*(79*c^6*d^
2*e^4 + 206*b*c^5*d*e^5 + b^2*c^4*e^6)*x^4 - 20*(271*c^6*d^3*e^3 - 683*b*c^5*d^2*e^4 - 19*b^2*c^4*d*e^5 + 2*b^
3*c^3*e^6)*x^3 - 3*(3169*c^6*d^4*e^2 - 3628*b*c^5*d^3*e^3 - 694*b^2*c^4*d^2*e^4 + 168*b^3*c^3*d*e^5 - 16*b^4*c
^2*e^6)*x^2 - 2*(1333*c^6*d^5*e + 1421*b*c^5*d^4*e^2 - 4142*b^2*c^4*d^3*e^3 + 1724*b^3*c^3*d^2*e^4 - 368*b^4*c
^2*d*e^5 + 32*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^5*e^2*x + c^5*d*e) - 2/45045*(3003*c^7*e^7
*x^7 + 12686*c^7*d^7 - 55754*b*c^6*d^6*e + 101034*b^2*c^5*d^5*e^2 - 97998*b^3*c^4*d^4*e^3 + 55296*b^4*c^3*d^3*
e^4 - 18336*b^5*c^2*d^2*e^5 + 3328*b^6*c*d*e^6 - 256*b^7*e^7 + 924*(11*c^7*d*e^6 + 4*b*c^6*e^7)*x^6 + 63*(111*
c^7*d^2*e^5 + 278*b*c^6*d*e^6 + b^2*c^5*e^7)*x^5 - 70*(175*c^7*d^3*e^4 - 453*b*c^6*d^2*e^5 - 9*b^2*c^5*d*e^6 +
b^3*c^4*e^7)*x^4 - 5*(4087*c^7*d^4*e^3 - 4900*b*c^6*d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^
4*c^3*e^7)*x^3 - 12*(542*c^7*d^5*e^2 + 11*b*c^6*d^4*e^3 - 862*b^2*c^5*d^3*e^4 + 389*b^3*c^4*d^2*e^5 - 88*b^4*c
^3*d*e^6 + 8*b^5*c^2*e^7)*x^2 + (6343*c^7*d^6*e - 21534*b*c^6*d^5*e^2 + 28983*b^2*c^5*d^4*e^3 - 20016*b^3*c^4*
d^3*e^4 + 7632*b^4*c^3*d^2*e^5 - 1536*b^5*c^2*d*e^6 + 128*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(
c^6*e^3*x + c^6*d*e^2)

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mupad [B]  time = 3.93, size = 863, normalized size = 2.06 \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 (16\,b\,e\,g+44\,c\,d\,g+15\,c\,e\,f\right )}{195}+\frac {2\,e^2\,x^5\,\sqrt {d+e\,x}\,\left (g\,b^2\,e^2+278\,g\,b\,c\,d\,e+70\,f\,b\,c\,e^2+111\,g\,c^2\,d^2+190\,f\,c^2\,d\,e\right )}{715\,c}+\frac {2\,c\,e^4\,g\,x^7\,\sqrt {d+e\,x}}{15}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-256\,g\,b^5\,e^5+2816\,g\,b^4\,c\,d\,e^4+384\,f\,b^4\,c\,e^5-12448\,g\,b^3\,c^2\,d^2\,e^3-4032\,f\,b^3\,c^2\,d\,e^4+27584\,g\,b^2\,c^3\,d^3\,e^2+16656\,f\,b^2\,c^3\,d^2\,e^3-30382\,g\,b\,c^4\,d^4\,e-33048\,f\,b\,c^4\,d^3\,e^2+12686\,g\,c^5\,d^5+29049\,f\,c^5\,d^4\,e\right )}{45045\,c^6\,e^3}+\frac {x^3\,\sqrt {d+e\,x}\,\left (160\,g\,b^4\,c^3\,e^7-1600\,g\,b^3\,c^4\,d\,e^6-240\,f\,b^3\,c^4\,e^7+6180\,g\,b^2\,c^5\,d^2\,e^5+2280\,f\,b^2\,c^5\,d\,e^6+49000\,g\,b\,c^6\,d^3\,e^4+81960\,f\,b\,c^6\,d^2\,e^5-40870\,g\,c^7\,d^4\,e^3-32520\,f\,c^7\,d^3\,e^4\right )}{45045\,c^6\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (-140\,g\,b^3\,c^4\,e^7+1260\,g\,b^2\,c^5\,d\,e^6+210\,f\,b^2\,c^5\,e^7+63420\,g\,b\,c^6\,d^2\,e^5+43260\,f\,b\,c^6\,d\,e^6-24500\,g\,c^7\,d^3\,e^4+16590\,f\,c^7\,d^2\,e^5\right )}{45045\,c^6\,e^3}-\frac {x^2\,\sqrt {d+e\,x}\,\left (192\,g\,b^5\,c^2\,e^7-2112\,g\,b^4\,c^3\,d\,e^6-288\,f\,b^4\,c^3\,e^7+9336\,g\,b^3\,c^4\,d^2\,e^5+3024\,f\,b^3\,c^4\,d\,e^6-20688\,g\,b^2\,c^5\,d^3\,e^4-12492\,f\,b^2\,c^5\,d^2\,e^5+264\,g\,b\,c^6\,d^4\,e^3-65304\,f\,b\,c^6\,d^3\,e^4+13008\,g\,c^7\,d^5\,e^2+57042\,f\,c^7\,d^4\,e^3\right )}{45045\,c^6\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (128\,g\,b^5\,e^5-1408\,g\,b^4\,c\,d\,e^4-192\,f\,b^4\,c\,e^5+6224\,g\,b^3\,c^2\,d^2\,e^3+2016\,f\,b^3\,c^2\,d\,e^4-13792\,g\,b^2\,c^3\,d^3\,e^2-8328\,f\,b^2\,c^3\,d^2\,e^3+15191\,g\,b\,c^4\,d^4\,e+16524\,f\,b\,c^4\,d^3\,e^2-6343\,g\,c^5\,d^5+7998\,f\,c^5\,d^4\,e\right )}{45045\,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)^(5/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),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)*(16*b*e*g + 44*c*d*g + 15*c*e*f))/19
5 + (2*e^2*x^5*(d + e*x)^(1/2)*(b^2*e^2*g + 111*c^2*d^2*g + 70*b*c*e^2*f + 190*c^2*d*e*f + 278*b*c*d*e*g))/(71
5*c) + (2*c*e^4*g*x^7*(d + e*x)^(1/2))/15 + (2*(b*e - c*d)^2*(d + e*x)^(1/2)*(12686*c^5*d^5*g - 256*b^5*e^5*g
+ 384*b^4*c*e^5*f + 29049*c^5*d^4*e*f - 30382*b*c^4*d^4*e*g + 2816*b^4*c*d*e^4*g - 33048*b*c^4*d^3*e^2*f - 403
2*b^3*c^2*d*e^4*f + 16656*b^2*c^3*d^2*e^3*f + 27584*b^2*c^3*d^3*e^2*g - 12448*b^3*c^2*d^2*e^3*g))/(45045*c^6*e
^3) + (x^3*(d + e*x)^(1/2)*(160*b^4*c^3*e^7*g - 240*b^3*c^4*e^7*f - 32520*c^7*d^3*e^4*f - 40870*c^7*d^4*e^3*g
+ 81960*b*c^6*d^2*e^5*f + 2280*b^2*c^5*d*e^6*f + 49000*b*c^6*d^3*e^4*g - 1600*b^3*c^4*d*e^6*g + 6180*b^2*c^5*d
^2*e^5*g))/(45045*c^6*e^3) + (x^4*(d + e*x)^(1/2)*(210*b^2*c^5*e^7*f - 140*b^3*c^4*e^7*g + 16590*c^7*d^2*e^5*f
- 24500*c^7*d^3*e^4*g + 43260*b*c^6*d*e^6*f + 63420*b*c^6*d^2*e^5*g + 1260*b^2*c^5*d*e^6*g))/(45045*c^6*e^3)
- (x^2*(d + e*x)^(1/2)*(192*b^5*c^2*e^7*g - 288*b^4*c^3*e^7*f + 57042*c^7*d^4*e^3*f + 13008*c^7*d^5*e^2*g - 65
304*b*c^6*d^3*e^4*f + 3024*b^3*c^4*d*e^6*f + 264*b*c^6*d^4*e^3*g - 2112*b^4*c^3*d*e^6*g - 12492*b^2*c^5*d^2*e^
5*f - 20688*b^2*c^5*d^3*e^4*g + 9336*b^3*c^4*d^2*e^5*g))/(45045*c^6*e^3) + (2*x*(b*e - c*d)*(d + e*x)^(1/2)*(1
28*b^5*e^5*g - 6343*c^5*d^5*g - 192*b^4*c*e^5*f + 7998*c^5*d^4*e*f + 15191*b*c^4*d^4*e*g - 1408*b^4*c*d*e^4*g
+ 16524*b*c^4*d^3*e^2*f + 2016*b^3*c^2*d*e^4*f - 8328*b^2*c^3*d^2*e^3*f - 13792*b^2*c^3*d^3*e^2*g + 6224*b^3*c
^2*d^2*e^3*g))/(45045*c^5*e^2)))/(x + d/e)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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