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

Optimal. Leaf size=694 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (91 \sqrt [3]{b} (4 a f+11 b c)-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (2 a g+13 b d)\right )}{10010 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a g+13 b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{182 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{27 a \sqrt{a+b x^3} (2 a g+13 b d)}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x} \]

[Out]

(27*a*c*Sqrt[a + b*x^3])/(10*x^2) - (27*a*d*Sqrt[a + b*x^3])/(7*x) + (27*a*(13*b
*d + 2*a*g)*Sqrt[a + b*x^3])/(91*b^(2/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) -
(2*a*Sqrt[a + b*x^3]*(27027*c*x - 19305*d*x^2 - 5005*e*x^3 - 2457*f*x^4 - 1485*g
*x^5))/(15015*x^3) + (2*(a + b*x^3)^(3/2)*(9009*c*x + 6435*d*x^2 + 5005*e*x^3 +
4095*f*x^4 + 3465*g*x^5))/(45045*x^3) - (2*a^(3/2)*e*ArcTanh[Sqrt[a + b*x^3]/Sqr
t[a]])/3 - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(13*b*d + 2*a*g)*(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]])/(182*b^(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]]*a*(91*b^(1/3)*(11*b*c + 4*a*f) - 110*(1 - Sqrt[3])*a^(1/3)*
(13*b*d + 2*a*g))*(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]])/(100
10*b^(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 = 1.68613, antiderivative size = 694, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (91 \sqrt [3]{b} (4 a f+11 b c)-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (2 a g+13 b d)\right )}{10010 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a g+13 b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{182 b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{27 a \sqrt{a+b x^3} (2 a g+13 b d)}{91 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 a \sqrt{a+b x^3} \left (27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right )}{15015 x^3}+\frac{2 \left (a+b x^3\right )^{3/2} \left (9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right )}{45045 x^3}+\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x} \]

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^3,x]

[Out]

(27*a*c*Sqrt[a + b*x^3])/(10*x^2) - (27*a*d*Sqrt[a + b*x^3])/(7*x) + (27*a*(13*b
*d + 2*a*g)*Sqrt[a + b*x^3])/(91*b^(2/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) -
(2*a*Sqrt[a + b*x^3]*(27027*c*x - 19305*d*x^2 - 5005*e*x^3 - 2457*f*x^4 - 1485*g
*x^5))/(15015*x^3) + (2*(a + b*x^3)^(3/2)*(9009*c*x + 6435*d*x^2 + 5005*e*x^3 +
4095*f*x^4 + 3465*g*x^5))/(45045*x^3) - (2*a^(3/2)*e*ArcTanh[Sqrt[a + b*x^3]/Sqr
t[a]])/3 - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(13*b*d + 2*a*g)*(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]])/(182*b^(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]]*a*(91*b^(1/3)*(11*b*c + 4*a*f) - 110*(1 - Sqrt[3])*a^(1/3)*
(13*b*d + 2*a*g))*(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]])/(100
10*b^(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**3,x)

[Out]

Timed out

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Mathematica [C]  time = 4.09528, size = 952, normalized size = 1.37 \[ \frac{\sqrt{b x^3+a} \left (4 b \left (9009 c+5 x \left (1287 d+7 x \left (99 g x^2+117 f x+143 e\right )\right )\right ) x^3+a \left (8 (10010 e+9 x (637 f+440 g x)) x^2-90090 d x-45045 c\right )\right )}{90090 x^2}-\frac{a \left (-17820 \sqrt{2} g \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}+29484 \sqrt [3]{b} f \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+20020 b^{2/3} e \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}-115830 \sqrt{2} b d \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}+81081 b^{4/3} c \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 )\right )}{30030 b^{2/3} \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^3,x]

[Out]

(Sqrt[a + b*x^3]*(a*(-45045*c - 90090*d*x + 8*x^2*(10010*e + 9*x*(637*f + 440*g*
x))) + 4*b*x^3*(9009*c + 5*x*(1287*d + 7*x*(143*e + 117*f*x + 99*g*x^2)))))/(900
90*x^2) - (a*(20020*Sqrt[a]*b^(2/3)*e*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]] + 81081
*b^(4/3)*c*((-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[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (
-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + 29484*a*b^(1/3)*f*((-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[ArcSin[S
qrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] -
115830*Sqrt[2]*a^(1/3)*b*d*((-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))]) - Ellipt
icF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (
-1)^(1/3))]) - 17820*Sqrt[2]*a^(4/3)*g*((-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))])))/(30030*b^(2/3)*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 = 0.012, size = 1613, normalized size = 2.3 \[ \text{result too large to display} \]

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^3,x)

[Out]

f*(2/11*b*x^4*(b*x^3+a)^(1/2)+28/55*a*x*(b*x^3+a)^(1/2)-18/55*I*a^2*3^(1/2)/b*(-
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)))+c*(-1/2*a*(b*x^3+a)^(1/2)/x^2+2/5*b*x*(b*x^3+a)^(1/2)-9/1
0*I*a*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)*Ellipt
icF(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*(-a*(b*x^3+a)^(1/2)/x+2/7*b*x^2*(b*x^3+
a)^(1/2)-9/7*I*a*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))))+e*(2/9*b*x^3*(b*x^3+a)^(1/2)+8/9*a*(b*x^3+a)^(1/2)-2/3*a^(3/2)*arctanh
((b*x^3+a)^(1/2)/a^(1/2)))+g*(2/13*b*x^5*(b*x^3+a)^(1/2)+32/91*a*x^2*(b*x^3+a)^(
1/2)-18/91*I*a^2*3^(1/2)/b*(-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))))

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

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^3,x, algorithm="maxima")

[Out]

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

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

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^3,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^3, x)

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Sympy [A]  time = 14.4792, size = 462, normalized size = 0.67 \[ \text{result too large to display} \]

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**3,x)

[Out]

a**(3/2)*c*gamma(-2/3)*hyper((-2/3, -1/2), (1/3,), b*x**3*exp_polar(I*pi)/a)/(3*
x**2*gamma(1/3)) + a**(3/2)*d*gamma(-1/3)*hyper((-1/2, -1/3), (2/3,), b*x**3*exp
_polar(I*pi)/a)/(3*x*gamma(2/3)) - 2*a**(3/2)*e*asinh(sqrt(a)/(sqrt(b)*x**(3/2))
)/3 + a**(3/2)*f*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/
a)/(3*gamma(4/3)) + a**(3/2)*g*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3
*exp_polar(I*pi)/a)/(3*gamma(5/3)) + sqrt(a)*b*c*x*gamma(1/3)*hyper((-1/2, 1/3),
(4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + sqrt(a)*b*d*x**2*gamma(2/3)*
hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + sqrt(a)*b*
f*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(
7/3)) + sqrt(a)*b*g*x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_polar(
I*pi)/a)/(3*gamma(8/3)) + 2*a**2*e/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2
*a*sqrt(b)*e*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + b*e*Piecewise((sqrt(a)*x**3/3,
Eq(b, 0)), (2*(a + b*x**3)**(3/2)/(9*b), True))

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

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^3,x, algorithm="giac")

[Out]

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