3.8.83 \(\int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [783]

Optimal. Leaf size=501 \[ \frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}} \]

[Out]

128/3465*(-a*e*g+c*d*f)^3*(10*a*e^2*g+c*d*(-11*d*g+e*f))*(2*a*e^2*g-c*d*(-d*g+3*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/c^6/d^6/e/g/(e*x+d)^(1/2)-32/1155*(-a*e*g+c*d*f)^2*(10*a*e^2*g+c*d*(-11*d*g+e*f))*(g*x+f)^2*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/g/(e*x+d)^(1/2)-16/693*(-a*e*g+c*d*f)*(10*a*e^2*g+c*d*(-11*d*g+e
*f))*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/g/(e*x+d)^(1/2)-2/99*(10*a*e^2*g+c*d*(-11*d*g+e
*f))*(g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/g/(e*x+d)^(1/2)+2/11*e*(g*x+f)^5*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/g/(e*x+d)^(1/2)-128/3465*(-a*e*g+c*d*f)^3*(10*a*e^2*g+c*d*(-11*d*g+e*f))*(e*x+d)
^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^5/d^5/e

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Rubi [A]
time = 0.55, antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {894, 884, 808, 662} \begin {gather*} \frac {128 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}-\frac {32 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(3465*c^6*d^6*e*g*Sqrt[d + e*x]) - (128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*
d*g))*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3465*c^5*d^5*e) - (32*(c*d*f - a*e*g)^2*(10*
a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(1155*c^4*d^4*g*Sqrt[d
+ e*x]) - (16*(c*d*f - a*e*g)*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(693*c^3*d^3*g*Sqrt[d + e*x]) - (2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^4*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2])/(99*c^2*d^2*g*Sqrt[d + e*x]) + (2*e*(f + g*x)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/(11*c*d*g*Sqrt[d + e*x])

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 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])

Rule 884

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

Rule 894

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(c*g*(n + p + 2))), x] - Dist[(b*e*g*(
n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x +
c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && Eq
Q[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {1}{11} \left (-11 d+\frac {10 a e^2}{c d}+\frac {e f}{g}\right ) \int \frac {\sqrt {d+e x} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (8 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{99 c^2 d^2 g}\\ &=-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (16 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{231 c^3 d^3 g}\\ &=-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (64 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1155 c^4 d^4 g}\\ &=-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}+\frac {\left (64 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3465 c^5 d^5 e g}\\ &=\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 380, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-1280 a^5 e^6 g^4+128 a^4 c d e^4 g^3 (44 e f+11 d g+5 e g x)-32 a^3 c^2 d^2 e^3 g^2 \left (22 d g (9 f+g x)+e \left (297 f^2+88 f g x+15 g^2 x^2\right )\right )+16 a^2 c^3 d^3 e^2 g \left (33 d g \left (21 f^2+6 f g x+g^2 x^2\right )+e \left (462 f^3+297 f^2 g x+132 f g^2 x^2+25 g^3 x^3\right )\right )-2 a c^4 d^4 e \left (44 d g \left (105 f^3+63 f^2 g x+27 f g^2 x^2+5 g^3 x^3\right )+e \left (1155 f^4+1848 f^3 g x+1782 f^2 g^2 x^2+880 f g^3 x^3+175 g^4 x^4\right )\right )+c^5 d^5 \left (11 d \left (315 f^4+420 f^3 g x+378 f^2 g^2 x^2+180 f g^3 x^3+35 g^4 x^4\right )+e x \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )\right )}{3465 c^6 d^6 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-1280*a^5*e^6*g^4 + 128*a^4*c*d*e^4*g^3*(44*e*f + 11*d*g + 5*e*g*x) - 32*a^3
*c^2*d^2*e^3*g^2*(22*d*g*(9*f + g*x) + e*(297*f^2 + 88*f*g*x + 15*g^2*x^2)) + 16*a^2*c^3*d^3*e^2*g*(33*d*g*(21
*f^2 + 6*f*g*x + g^2*x^2) + e*(462*f^3 + 297*f^2*g*x + 132*f*g^2*x^2 + 25*g^3*x^3)) - 2*a*c^4*d^4*e*(44*d*g*(1
05*f^3 + 63*f^2*g*x + 27*f*g^2*x^2 + 5*g^3*x^3) + e*(1155*f^4 + 1848*f^3*g*x + 1782*f^2*g^2*x^2 + 880*f*g^3*x^
3 + 175*g^4*x^4)) + c^5*d^5*(11*d*(315*f^4 + 420*f^3*g*x + 378*f^2*g^2*x^2 + 180*f*g^3*x^3 + 35*g^4*x^4) + e*x
*(1155*f^4 + 2772*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4*x^4))))/(3465*c^6*d^6*Sqrt[d + e*x])

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Maple [A]
time = 0.12, size = 623, normalized size = 1.24

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right )}{3465 \sqrt {e x +d}\, c^{6} d^{6}}\)	\(623\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right ) \sqrt {e x +d}}{3465 c^{6} d^{6} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(641\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-315*c^5*d^5*e*g^4*x^5+350*a*c^4*d^4*e^2*g^4*x^4-385*c^5*d^
6*g^4*x^4-1540*c^5*d^5*e*f*g^3*x^4-400*a^2*c^3*d^3*e^3*g^4*x^3+440*a*c^4*d^5*e*g^4*x^3+1760*a*c^4*d^4*e^2*f*g^
3*x^3-1980*c^5*d^6*f*g^3*x^3-2970*c^5*d^5*e*f^2*g^2*x^3+480*a^3*c^2*d^2*e^4*g^4*x^2-528*a^2*c^3*d^4*e^2*g^4*x^
2-2112*a^2*c^3*d^3*e^3*f*g^3*x^2+2376*a*c^4*d^5*e*f*g^3*x^2+3564*a*c^4*d^4*e^2*f^2*g^2*x^2-4158*c^5*d^6*f^2*g^
2*x^2-2772*c^5*d^5*e*f^3*g*x^2-640*a^4*c*d*e^5*g^4*x+704*a^3*c^2*d^3*e^3*g^4*x+2816*a^3*c^2*d^2*e^4*f*g^3*x-31
68*a^2*c^3*d^4*e^2*f*g^3*x-4752*a^2*c^3*d^3*e^3*f^2*g^2*x+5544*a*c^4*d^5*e*f^2*g^2*x+3696*a*c^4*d^4*e^2*f^3*g*
x-4620*c^5*d^6*f^3*g*x-1155*c^5*d^5*e*f^4*x+1280*a^5*e^6*g^4-1408*a^4*c*d^2*e^4*g^4-5632*a^4*c*d*e^5*f*g^3+633
6*a^3*c^2*d^3*e^3*f*g^3+9504*a^3*c^2*d^2*e^4*f^2*g^2-11088*a^2*c^3*d^4*e^2*f^2*g^2-7392*a^2*c^3*d^3*e^3*f^3*g+
9240*a*c^4*d^5*e*f^3*g+2310*a*c^4*d^4*e^2*f^4-3465*c^5*d^6*f^4)/c^6/d^6

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Maxima [A]
time = 0.37, size = 678, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} e + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{4}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {8 \, {\left (3 \, c^{3} d^{3} x^{3} e - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{3} g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {4 \, {\left (15 \, c^{4} d^{4} x^{4} e + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f^{2} g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {8 \, {\left (35 \, c^{5} d^{5} x^{5} e - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} f g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} + \frac {2 \, {\left (315 \, c^{6} d^{6} x^{6} e + 1408 \, a^{5} c d^{2} e^{5} - 1280 \, a^{6} e^{7} + 35 \, {\left (11 \, c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{5} - 5 \, {\left (11 \, a c^{5} d^{6} e - 10 \, a^{2} c^{4} d^{4} e^{3}\right )} x^{4} + 8 \, {\left (11 \, a^{2} c^{4} d^{5} e^{2} - 10 \, a^{3} c^{3} d^{3} e^{4}\right )} x^{3} - 16 \, {\left (11 \, a^{3} c^{3} d^{4} e^{3} - 10 \, a^{4} c^{2} d^{2} e^{5}\right )} x^{2} + 64 \, {\left (11 \, a^{4} c^{2} d^{3} e^{4} - 10 \, a^{5} c d e^{6}\right )} x\right )} g^{4}}{3465 \, \sqrt {c d x + a e} c^{6} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/3*(c^2*d^2*x^2*e + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)*f^4/(sqrt(c*d*x + a*e)*c^2*d^2) + 8/
15*(3*c^3*d^3*x^3*e - 10*a^2*c*d^2*e^2 + 8*a^3*e^4 + (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*
c*d*e^3)*x)*f^3*g/(sqrt(c*d*x + a*e)*c^3*d^3) + 4/35*(15*c^4*d^4*x^4*e + 56*a^3*c*d^2*e^3 - 48*a^4*e^5 + 3*(7*
c^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*x^2 + 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)
*x)*f^2*g^2/(sqrt(c*d*x + a*e)*c^4*d^4) + 8/315*(35*c^5*d^5*x^5*e - 144*a^4*c*d^2*e^4 + 128*a^5*e^6 + 5*(9*c^5
*d^6 - a*c^4*d^4*e^2)*x^4 - (9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*x^3 + 2*(9*a^2*c^3*d^4*e^2 - 8*a^3*c^2*d^2*e^4
)*x^2 - 8*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*x)*f*g^3/(sqrt(c*d*x + a*e)*c^5*d^5) + 2/3465*(315*c^6*d^6*x^6*e
+ 1408*a^5*c*d^2*e^5 - 1280*a^6*e^7 + 35*(11*c^6*d^7 - a*c^5*d^5*e^2)*x^5 - 5*(11*a*c^5*d^6*e - 10*a^2*c^4*d^
4*e^3)*x^4 + 8*(11*a^2*c^4*d^5*e^2 - 10*a^3*c^3*d^3*e^4)*x^3 - 16*(11*a^3*c^3*d^4*e^3 - 10*a^4*c^2*d^2*e^5)*x^
2 + 64*(11*a^4*c^2*d^3*e^4 - 10*a^5*c*d*e^6)*x)*g^4/(sqrt(c*d*x + a*e)*c^6*d^6)

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Fricas [A]
time = 3.85, size = 608, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (385 \, c^{5} d^{6} g^{4} x^{4} + 1980 \, c^{5} d^{6} f g^{3} x^{3} + 4158 \, c^{5} d^{6} f^{2} g^{2} x^{2} + 4620 \, c^{5} d^{6} f^{3} g x + 3465 \, c^{5} d^{6} f^{4} - 1280 \, a^{5} g^{4} e^{6} + 128 \, {\left (5 \, a^{4} c d g^{4} x + 44 \, a^{4} c d f g^{3}\right )} e^{5} - 32 \, {\left (15 \, a^{3} c^{2} d^{2} g^{4} x^{2} + 88 \, a^{3} c^{2} d^{2} f g^{3} x + 297 \, a^{3} c^{2} d^{2} f^{2} g^{2} - 44 \, a^{4} c d^{2} g^{4}\right )} e^{4} + 16 \, {\left (25 \, a^{2} c^{3} d^{3} g^{4} x^{3} + 132 \, a^{2} c^{3} d^{3} f g^{3} x^{2} + 462 \, a^{2} c^{3} d^{3} f^{3} g - 396 \, a^{3} c^{2} d^{3} f g^{3} + 11 \, {\left (27 \, a^{2} c^{3} d^{3} f^{2} g^{2} - 4 \, a^{3} c^{2} d^{3} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (175 \, a c^{4} d^{4} g^{4} x^{4} + 880 \, a c^{4} d^{4} f g^{3} x^{3} + 1155 \, a c^{4} d^{4} f^{4} - 5544 \, a^{2} c^{3} d^{4} f^{2} g^{2} + 66 \, {\left (27 \, a c^{4} d^{4} f^{2} g^{2} - 4 \, a^{2} c^{3} d^{4} g^{4}\right )} x^{2} + 264 \, {\left (7 \, a c^{4} d^{4} f^{3} g - 6 \, a^{2} c^{3} d^{4} f g^{3}\right )} x\right )} e^{2} + {\left (315 \, c^{5} d^{5} g^{4} x^{5} + 1540 \, c^{5} d^{5} f g^{3} x^{4} - 9240 \, a c^{4} d^{5} f^{3} g + 110 \, {\left (27 \, c^{5} d^{5} f^{2} g^{2} - 4 \, a c^{4} d^{5} g^{4}\right )} x^{3} + 396 \, {\left (7 \, c^{5} d^{5} f^{3} g - 6 \, a c^{4} d^{5} f g^{3}\right )} x^{2} + 231 \, {\left (5 \, c^{5} d^{5} f^{4} - 24 \, a c^{4} d^{5} f^{2} g^{2}\right )} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3465 \, {\left (c^{6} d^{6} x e + c^{6} d^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(385*c^5*d^6*g^4*x^4 + 1980*c^5*d^6*f*g^3*x^3 + 4158*c^5*d^6*f^2*g^2*x^2 + 4620*c^5*d^6*f^3*g*x + 3465*
c^5*d^6*f^4 - 1280*a^5*g^4*e^6 + 128*(5*a^4*c*d*g^4*x + 44*a^4*c*d*f*g^3)*e^5 - 32*(15*a^3*c^2*d^2*g^4*x^2 + 8
8*a^3*c^2*d^2*f*g^3*x + 297*a^3*c^2*d^2*f^2*g^2 - 44*a^4*c*d^2*g^4)*e^4 + 16*(25*a^2*c^3*d^3*g^4*x^3 + 132*a^2
*c^3*d^3*f*g^3*x^2 + 462*a^2*c^3*d^3*f^3*g - 396*a^3*c^2*d^3*f*g^3 + 11*(27*a^2*c^3*d^3*f^2*g^2 - 4*a^3*c^2*d^
3*g^4)*x)*e^3 - 2*(175*a*c^4*d^4*g^4*x^4 + 880*a*c^4*d^4*f*g^3*x^3 + 1155*a*c^4*d^4*f^4 - 5544*a^2*c^3*d^4*f^2
*g^2 + 66*(27*a*c^4*d^4*f^2*g^2 - 4*a^2*c^3*d^4*g^4)*x^2 + 264*(7*a*c^4*d^4*f^3*g - 6*a^2*c^3*d^4*f*g^3)*x)*e^
2 + (315*c^5*d^5*g^4*x^5 + 1540*c^5*d^5*f*g^3*x^4 - 9240*a*c^4*d^5*f^3*g + 110*(27*c^5*d^5*f^2*g^2 - 4*a*c^4*d
^5*g^4)*x^3 + 396*(7*c^5*d^5*f^3*g - 6*a*c^4*d^5*f*g^3)*x^2 + 231*(5*c^5*d^5*f^4 - 24*a*c^4*d^5*f^2*g^2)*x)*e)
*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^6*d^6*x*e + c^6*d^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{4}}{\sqrt {\left (d + e x\right ) \left (a e + c d 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)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral((d + e*x)**(3/2)*(f + g*x)**4/sqrt((d + e*x)*(a*e + c*d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1786 vs. \(2 (483) = 966\).
time = 3.95, size = 1786, normalized size = 3.56 \begin {gather*} \frac {2 \, {\left (c^{5} d^{6} f^{4} - 4 \, a c^{4} d^{5} f^{3} g e - a c^{4} d^{4} f^{4} e^{2} + 6 \, a^{2} c^{3} d^{4} f^{2} g^{2} e^{2} + 4 \, a^{2} c^{3} d^{3} f^{3} g e^{3} - 4 \, a^{3} c^{2} d^{3} f g^{3} e^{3} - 6 \, a^{3} c^{2} d^{2} f^{2} g^{2} e^{4} + a^{4} c d^{2} g^{4} e^{4} + 4 \, a^{4} c d f g^{3} e^{5} - a^{5} g^{4} e^{6}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{6} d^{6}} - \frac {4 \, {\left (35 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{10} g^{4} - 220 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{9} f g^{3} e + 594 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{8} f^{2} g^{2} e^{2} + 45 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{8} g^{4} e^{2} - 924 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{7} f^{3} g e^{3} - 308 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{7} f g^{3} e^{3} + 1155 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{6} f^{4} e^{4} + 990 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{6} f^{2} g^{2} e^{4} + 64 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{6} g^{4} e^{4} - 2772 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{5} f^{3} g e^{5} - 528 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{5} f g^{3} e^{5} - 1155 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{4} f^{4} e^{6} + 3168 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{4} f^{2} g^{2} e^{6} + 112 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c^{2} d^{4} g^{4} e^{6} + 3696 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} f^{3} g e^{7} - 1760 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c^{2} d^{3} f g^{3} e^{7} - 4752 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c^{2} d^{2} f^{2} g^{2} e^{8} + 384 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} c d^{2} g^{4} e^{8} + 2816 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} c d f g^{3} e^{9} - 640 \, \sqrt {-c d^{2} e + a e^{3}} a^{5} g^{4} e^{10}\right )} e^{\left (-5\right )}}{3465 \, c^{6} d^{6}} + \frac {2 \, {\left (4620 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{5} f^{3} g e^{7} + 1155 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{4} f^{4} e^{8} - 13860 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{4} f^{2} g^{2} e^{8} + 4158 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{4} f^{2} g^{2} e^{5} - 9240 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} f^{3} g e^{9} + 13860 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{2} d^{3} f g^{3} e^{9} + 2772 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{3} f^{3} g e^{6} - 8316 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{2} d^{3} f g^{3} e^{6} + 1980 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{2} d^{3} f g^{3} e^{3} + 20790 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{2} d^{2} f^{2} g^{2} e^{10} - 4620 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} c d^{2} g^{4} e^{10} - 12474 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{2} d^{2} f^{2} g^{2} e^{7} + 4158 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} c d^{2} g^{4} e^{7} + 2970 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{2} d^{2} f^{2} g^{2} e^{4} - 1980 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a c d^{2} g^{4} e^{4} + 385 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} c d^{2} g^{4} e - 18480 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} c d f g^{3} e^{11} + 16632 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} c d f g^{3} e^{8} - 7920 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a c d f g^{3} e^{5} + 1540 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} c d f g^{3} e^{2} + 5775 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{4} g^{4} e^{12} - 6930 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{3} g^{4} e^{9} + 4950 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a^{2} g^{4} e^{6} - 1925 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} a g^{4} e^{3} + 315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {11}{2}} g^{4}\right )} e^{\left (-10\right )}}{3465 \, c^{6} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

2*(c^5*d^6*f^4 - 4*a*c^4*d^5*f^3*g*e - a*c^4*d^4*f^4*e^2 + 6*a^2*c^3*d^4*f^2*g^2*e^2 + 4*a^2*c^3*d^3*f^3*g*e^3
- 4*a^3*c^2*d^3*f*g^3*e^3 - 6*a^3*c^2*d^2*f^2*g^2*e^4 + a^4*c*d^2*g^4*e^4 + 4*a^4*c*d*f*g^3*e^5 - a^5*g^4*e^6
)*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*e^(-1)/(c^6*d^6) - 4/3465*(35*sqrt(-c*d^2*e + a*e^3)*c^5*d^10*g^4 -
220*sqrt(-c*d^2*e + a*e^3)*c^5*d^9*f*g^3*e + 594*sqrt(-c*d^2*e + a*e^3)*c^5*d^8*f^2*g^2*e^2 + 45*sqrt(-c*d^2*e
+ a*e^3)*a*c^4*d^8*g^4*e^2 - 924*sqrt(-c*d^2*e + a*e^3)*c^5*d^7*f^3*g*e^3 - 308*sqrt(-c*d^2*e + a*e^3)*a*c^4*
d^7*f*g^3*e^3 + 1155*sqrt(-c*d^2*e + a*e^3)*c^5*d^6*f^4*e^4 + 990*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^6*f^2*g^2*e^4
+ 64*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*g^4*e^4 - 2772*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^5*f^3*g*e^5 - 528*sqrt(
-c*d^2*e + a*e^3)*a^2*c^3*d^5*f*g^3*e^5 - 1155*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^4*f^4*e^6 + 3168*sqrt(-c*d^2*e +
a*e^3)*a^2*c^3*d^4*f^2*g^2*e^6 + 112*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*g^4*e^6 + 3696*sqrt(-c*d^2*e + a*e^3)
*a^2*c^3*d^3*f^3*g*e^7 - 1760*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^3*f*g^3*e^7 - 4752*sqrt(-c*d^2*e + a*e^3)*a^3*c
^2*d^2*f^2*g^2*e^8 + 384*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*g^4*e^8 + 2816*sqrt(-c*d^2*e + a*e^3)*a^4*c*d*f*g^3*
e^9 - 640*sqrt(-c*d^2*e + a*e^3)*a^5*g^4*e^10)*e^(-5)/(c^6*d^6) + 2/3465*(4620*((x*e + d)*c*d*e - c*d^2*e + a*
e^3)^(3/2)*c^4*d^5*f^3*g*e^7 + 1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^4*d^4*f^4*e^8 - 13860*((x*e +
d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^3*d^4*f^2*g^2*e^8 + 4158*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^3*d
^4*f^2*g^2*e^5 - 9240*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^3*d^3*f^3*g*e^9 + 13860*((x*e + d)*c*d*e -
c*d^2*e + a*e^3)^(3/2)*a^2*c^2*d^3*f*g^3*e^9 + 2772*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^3*d^3*f^3*g*e
^6 - 8316*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^2*d^3*f*g^3*e^6 + 1980*((x*e + d)*c*d*e - c*d^2*e + a*
e^3)^(7/2)*c^2*d^3*f*g^3*e^3 + 20790*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^2*d^2*f^2*g^2*e^10 - 4620
*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*c*d^2*g^4*e^10 - 12474*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2
)*a*c^2*d^2*f^2*g^2*e^7 + 4158*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c*d^2*g^4*e^7 + 2970*((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(7/2)*c^2*d^2*f^2*g^2*e^4 - 1980*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*c*d^2*g^4
*e^4 + 385*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*c*d^2*g^4*e - 18480*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(
3/2)*a^3*c*d*f*g^3*e^11 + 16632*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c*d*f*g^3*e^8 - 7920*((x*e + d)*
c*d*e - c*d^2*e + a*e^3)^(7/2)*a*c*d*f*g^3*e^5 + 1540*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*c*d*f*g^3*e^2
+ 5775*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*g^4*e^12 - 6930*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)
*a^3*g^4*e^9 + 4950*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*g^4*e^6 - 1925*((x*e + d)*c*d*e - c*d^2*e +
a*e^3)^(9/2)*a*g^4*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2)*g^4)*e^(-10)/(c^6*d^6)

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Mupad [B]
time = 4.09, size = 653, normalized size = 1.30 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5\,\sqrt {d+e\,x}}{11\,c\,d}-\frac {\sqrt {d+e\,x}\,\left (2560\,a^5\,e^6\,g^4-2816\,a^4\,c\,d^2\,e^4\,g^4-11264\,a^4\,c\,d\,e^5\,f\,g^3+12672\,a^3\,c^2\,d^3\,e^3\,f\,g^3+19008\,a^3\,c^2\,d^2\,e^4\,f^2\,g^2-22176\,a^2\,c^3\,d^4\,e^2\,f^2\,g^2-14784\,a^2\,c^3\,d^3\,e^3\,f^3\,g+18480\,a\,c^4\,d^5\,e\,f^3\,g+4620\,a\,c^4\,d^4\,e^2\,f^4-6930\,c^5\,d^6\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (1280\,a^4\,c\,d\,e^5\,g^4-1408\,a^3\,c^2\,d^3\,e^3\,g^4-5632\,a^3\,c^2\,d^2\,e^4\,f\,g^3+6336\,a^2\,c^3\,d^4\,e^2\,f\,g^3+9504\,a^2\,c^3\,d^3\,e^3\,f^2\,g^2-11088\,a\,c^4\,d^5\,e\,f^2\,g^2-7392\,a\,c^4\,d^4\,e^2\,f^3\,g+9240\,c^5\,d^6\,f^3\,g+2310\,c^5\,d^5\,e\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,a^3\,c^2\,d^2\,e^4\,g^4+1056\,a^2\,c^3\,d^4\,e^2\,g^4+4224\,a^2\,c^3\,d^3\,e^3\,f\,g^3-4752\,a\,c^4\,d^5\,e\,f\,g^3-7128\,a\,c^4\,d^4\,e^2\,f^2\,g^2+8316\,c^5\,d^6\,f^2\,g^2+5544\,c^5\,d^5\,e\,f^3\,g\right )}{3465\,c^6\,d^6\,e}+\frac {4\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (40\,a^2\,e^3\,g^2-44\,a\,c\,d^2\,e\,g^2-176\,a\,c\,d\,e^2\,f\,g+198\,c^2\,d^3\,f\,g+297\,c^2\,d^2\,e\,f^2\right )}{693\,c^3\,d^3\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}\,\left (11\,c\,g\,d^2+44\,c\,f\,d\,e-10\,a\,g\,e^2\right )}{99\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^4*(d + e*x)^(3/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*g^4*x^5*(d + e*x)^(1/2))/(11*c*d) - ((d + e*x)^(1/2)*(2560*
a^5*e^6*g^4 - 6930*c^5*d^6*f^4 + 4620*a*c^4*d^4*e^2*f^4 - 2816*a^4*c*d^2*e^4*g^4 - 14784*a^2*c^3*d^3*e^3*f^3*g
+ 12672*a^3*c^2*d^3*e^3*f*g^3 + 18480*a*c^4*d^5*e*f^3*g - 11264*a^4*c*d*e^5*f*g^3 - 22176*a^2*c^3*d^4*e^2*f^2
*g^2 + 19008*a^3*c^2*d^2*e^4*f^2*g^2))/(3465*c^6*d^6*e) + (x*(d + e*x)^(1/2)*(2310*c^5*d^5*e*f^4 + 9240*c^5*d^
6*f^3*g - 1408*a^3*c^2*d^3*e^3*g^4 + 1280*a^4*c*d*e^5*g^4 - 7392*a*c^4*d^4*e^2*f^3*g - 11088*a*c^4*d^5*e*f^2*g
^2 + 6336*a^2*c^3*d^4*e^2*f*g^3 - 5632*a^3*c^2*d^2*e^4*f*g^3 + 9504*a^2*c^3*d^3*e^3*f^2*g^2))/(3465*c^6*d^6*e)
+ (x^2*(d + e*x)^(1/2)*(8316*c^5*d^6*f^2*g^2 + 1056*a^2*c^3*d^4*e^2*g^4 - 960*a^3*c^2*d^2*e^4*g^4 + 5544*c^5*
d^5*e*f^3*g - 7128*a*c^4*d^4*e^2*f^2*g^2 + 4224*a^2*c^3*d^3*e^3*f*g^3 - 4752*a*c^4*d^5*e*f*g^3))/(3465*c^6*d^6
*e) + (4*g^2*x^3*(d + e*x)^(1/2)*(40*a^2*e^3*g^2 + 297*c^2*d^2*e*f^2 + 198*c^2*d^3*f*g - 44*a*c*d^2*e*g^2 - 17
6*a*c*d*e^2*f*g))/(693*c^3*d^3*e) + (2*g^3*x^4*(d + e*x)^(1/2)*(11*c*d^2*g - 10*a*e^2*g + 44*c*d*e*f))/(99*c^2
*d^2*e)))/(x + d/e)

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