3.457 \(\int \frac{\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^8} \, dx\)

Optimal. Leaf size=746 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (14 a f+b c)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{560 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a f+b c) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a f+b c)}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{b \sqrt{a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac{1}{420} \left (a+b x^3\right )^{3/2} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right )-\frac{27 b \sqrt{a+b x^3} (14 a f+b c)}{112 a x}+\frac{27 b c \sqrt{a+b x^3}}{280 x^4}-\frac{b (4 a g+b d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{b d \sqrt{a+b x^3}}{4 x^3}+\frac{27 b e \sqrt{a+b x^3}}{20 x^2} \]

[Out]

(27*b*c*Sqrt[a + b*x^3])/(280*x^4) + (b*d*Sqrt[a + b*x^3])/(4*x^3) + (27*b*e*Sqr
t[a + b*x^3])/(20*x^2) - (27*b*(b*c + 14*a*f)*Sqrt[a + b*x^3])/(112*a*x) + (27*b
^(4/3)*(b*c + 14*a*f)*Sqrt[a + b*x^3])/(112*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x
)) - (((60*c)/x^7 + (70*d)/x^6 + (84*e)/x^5 + (105*f)/x^4 + (140*g)/x^3)*(a + b*
x^3)^(3/2))/420 - (b*Sqrt[a + b*x^3]*(36*c*x + 70*d*x^2 + 252*e*x^3 - 315*f*x^4
- 140*g*x^5))/(140*x^5) - (b*(b*d + 4*a*g)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*
Sqrt[a]) - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(4/3)*(b*c + 14*a*f)*(a^(1/3) + b^(1/
3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3]
)*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(224*a^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) +
b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]) + (9*3^(3/4)
*Sqrt[2 + Sqrt[3]]*b^(4/3)*(28*a^(2/3)*b^(1/3)*e - 5*(1 - Sqrt[3])*(b*c + 14*a*f
))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/
3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(560*a^(2/3)*Sqrt[(
a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b
*x^3])

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Rubi [A]  time = 2.37261, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt{3}\right ) (14 a f+b c)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{560 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a f+b c) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a f+b c)}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{b \sqrt{a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac{1}{420} \left (a+b x^3\right )^{3/2} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right )-\frac{27 b \sqrt{a+b x^3} (14 a f+b c)}{112 a x}+\frac{27 b c \sqrt{a+b x^3}}{280 x^4}-\frac{b (4 a g+b d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{b d \sqrt{a+b x^3}}{4 x^3}+\frac{27 b e \sqrt{a+b x^3}}{20 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^8,x]

[Out]

(27*b*c*Sqrt[a + b*x^3])/(280*x^4) + (b*d*Sqrt[a + b*x^3])/(4*x^3) + (27*b*e*Sqr
t[a + b*x^3])/(20*x^2) - (27*b*(b*c + 14*a*f)*Sqrt[a + b*x^3])/(112*a*x) + (27*b
^(4/3)*(b*c + 14*a*f)*Sqrt[a + b*x^3])/(112*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x
)) - (((60*c)/x^7 + (70*d)/x^6 + (84*e)/x^5 + (105*f)/x^4 + (140*g)/x^3)*(a + b*
x^3)^(3/2))/420 - (b*Sqrt[a + b*x^3]*(36*c*x + 70*d*x^2 + 252*e*x^3 - 315*f*x^4
- 140*g*x^5))/(140*x^5) - (b*(b*d + 4*a*g)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*
Sqrt[a]) - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(4/3)*(b*c + 14*a*f)*(a^(1/3) + b^(1/
3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3]
)*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(224*a^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) +
b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]) + (9*3^(3/4)
*Sqrt[2 + Sqrt[3]]*b^(4/3)*(28*a^(2/3)*b^(1/3)*e - 5*(1 - Sqrt[3])*(b*c + 14*a*f
))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/
3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(560*a^(2/3)*Sqrt[(
a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b
*x^3])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**8,x)

[Out]

Timed out

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Mathematica [C]  time = 4.16604, size = 897, normalized size = 1.2 \[ -\frac{\sqrt{b x^3+a} \left (405 b^2 c x^6+2 a b \left (255 c+7 x \left (50 d+x \left (-80 g x^2+165 f x+78 e\right )\right )\right ) x^3+4 a^2 (60 c+7 x (10 d+x (12 e+5 x (3 f+4 g x))))\right )}{1680 a x^7}-\frac{b \left (560 g \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a} \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) a^{3/2}-1890 \sqrt{2} \sqrt [3]{b} f \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{3 i+\sqrt{3}}} \left (-\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )-F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right ) a^{4/3}+756 b^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right ) a+140 b d \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a} \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) \sqrt{a}-135 \sqrt{2} b^{4/3} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{3 i+\sqrt{3}}} \left (-\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )-F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right ) \sqrt [3]{a}\right )}{560 a \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^8,x]

[Out]

-(Sqrt[a + b*x^3]*(405*b^2*c*x^6 + 2*a*b*x^3*(255*c + 7*x*(50*d + x*(78*e + 165*
f*x - 80*g*x^2))) + 4*a^2*(60*c + 7*x*(10*d + x*(12*e + 5*x*(3*f + 4*g*x))))))/(
1680*a*x^7) - (b*(140*Sqrt[a]*b*d*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-
1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]] + 560*a^(3/
2)*g*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a +
b*x^3]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]] + 756*a*b^(2/3)*e*((-1)^(1/3)*a^(1/3) -
b^(1/3)*x)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[((-1)^(1/
3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSi
n[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)
] - 135*Sqrt[2]*a^(1/3)*b^(4/3)*c*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-1)^(1
/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[(I*(1 + (
b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt
[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) -
EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(
-1 + (-1)^(1/3))]) - 1890*Sqrt[2]*a^(4/3)*b^(1/3)*f*((-1)^(1/3)*a^(1/3) - b^(1/3
)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3
))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)^(2/3))*Ell
ipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1
+ (-1)^(1/3))]) - EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1
/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))])))/(560*a*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)
*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])

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Maple [B]  time = 0.015, size = 1375, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^8,x)

[Out]

c*(-1/7*a*(b*x^3+a)^(1/2)/x^7-17/56*b*(b*x^3+a)^(1/2)/x^4-27/112/a*b^2*(b*x^3+a)
^(1/2)/x-9/112*I/a*b^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3
^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))
/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*
b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^
3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3
*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-
a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(
1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/
2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/
2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(
1/3)))^(1/2))))+d*(-1/6*a*(b*x^3+a)^(1/2)/x^6-5/12*b*(b*x^3+a)^(1/2)/x^3-1/4*b^2
*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2))+e*(-1/5*a*(b*x^3+a)^(1/2)/x^5-13/20*b
*(b*x^3+a)^(1/2)/x^2-9/20*I*b*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-
1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)
^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2
/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2
)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)
/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(
-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))+f*(-1/4*a*(b*x^3+
a)^(1/2)/x^4-11/8*b*(b*x^3+a)^(1/2)/x-9/8*I*b*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b
*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*
((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^
(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*
b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^
2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b
^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(
-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*Ellipti
cF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2
)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2
*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))+g*(-1/3*a*(b*x^3+a)^(1/2)/x^3+2/3*b*(b*x^
3+a)^(1/2)-a^(1/2)*b*arctanh((b*x^3+a)^(1/2)/a^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^8,x, algorithm="maxima")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^8, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b g x^{7} + b f x^{6} + b e x^{5} +{\left (b d + a g\right )} x^{4} + a e x^{2} +{\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt{b x^{3} + a}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^8,x, algorithm="fricas")

[Out]

integral((b*g*x^7 + b*f*x^6 + b*e*x^5 + (b*d + a*g)*x^4 + a*e*x^2 + (b*c + a*f)*
x^3 + a*d*x + a*c)*sqrt(b*x^3 + a)/x^8, x)

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Sympy [A]  time = 25.2408, size = 536, normalized size = 0.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**8,x)

[Out]

a**(3/2)*c*gamma(-7/3)*hyper((-7/3, -1/2), (-4/3,), b*x**3*exp_polar(I*pi)/a)/(3
*x**7*gamma(-4/3)) + a**(3/2)*e*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*
exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + a**(3/2)*f*gamma(-4/3)*hyper((-4/3, -1
/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3)) + sqrt(a)*b*c*gamma
(-4/3)*hyper((-4/3, -1/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3
)) + sqrt(a)*b*e*gamma(-2/3)*hyper((-2/3, -1/2), (1/3,), b*x**3*exp_polar(I*pi)/
a)/(3*x**2*gamma(1/3)) + sqrt(a)*b*f*gamma(-1/3)*hyper((-1/2, -1/3), (2/3,), b*x
**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3)) - sqrt(a)*b*g*asinh(sqrt(a)/(sqrt(b)*x**
(3/2))) - a**2*d/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*d/(4*x**
(9/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*g*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) + 2*
a*sqrt(b)*g/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**(3/2)*d*sqrt(a/(b*x**3) + 1)/
(3*x**(3/2)) - b**(3/2)*d/(12*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*b**(3/2)*g*x**(
3/2)/(3*sqrt(a/(b*x**3) + 1)) - b**2*d*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*sqrt
(a))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^8,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^8, x)