3.418 \(\int \frac{x^3 \left (c+d x+e x^2\right )}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=583 \[ -\frac{4 \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 a d \sqrt{a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b} \]

[Out]

(-4*a*e*Sqrt[a + b*x^3])/(9*b^2) + (2*c*x*Sqrt[a + b*x^3])/(5*b) + (2*d*x^2*Sqrt
[a + b*x^3])/(7*b) + (2*e*x^3*Sqrt[a + b*x^3])/(9*b) - (8*a*d*Sqrt[a + b*x^3])/(
7*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^
(4/3)*d*(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]])/(7*b^(5/3)*Sqr
t[(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]) - (4*Sqrt[2 + Sqrt[3]]*a*(7*b^(1/3)*c - 10*(1 - Sqrt[3])*a^(1/3)*d)*(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]])/(35*3^(1/4)*b^(5/3)*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 = 1.33722, antiderivative size = 583, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{4 \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 a d \sqrt{a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(c + d*x + e*x^2))/Sqrt[a + b*x^3],x]

[Out]

(-4*a*e*Sqrt[a + b*x^3])/(9*b^2) + (2*c*x*Sqrt[a + b*x^3])/(5*b) + (2*d*x^2*Sqrt
[a + b*x^3])/(7*b) + (2*e*x^3*Sqrt[a + b*x^3])/(9*b) - (8*a*d*Sqrt[a + b*x^3])/(
7*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^
(4/3)*d*(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]])/(7*b^(5/3)*Sqr
t[(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]) - (4*Sqrt[2 + Sqrt[3]]*a*(7*b^(1/3)*c - 10*(1 - Sqrt[3])*a^(1/3)*d)*(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]])/(35*3^(1/4)*b^(5/3)*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 [A]  time = 163.345, size = 522, normalized size = 0.9 \[ \frac{4 \sqrt [4]{3} a^{\frac{4}{3}} d \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 b^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{4 a e \sqrt{a + b x^{3}}}{9 b^{2}} - \frac{8 a d \sqrt{a + b x^{3}}}{7 b^{\frac{5}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} - \frac{4 \cdot 3^{\frac{3}{4}} a \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (- 10 \sqrt [3]{a} d \left (- \sqrt{3} + 1\right ) + 7 \sqrt [3]{b} c\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{105 b^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{2 c x \sqrt{a + b x^{3}}}{5 b} + \frac{2 d x^{2} \sqrt{a + b x^{3}}}{7 b} + \frac{2 e x^{3} \sqrt{a + b x^{3}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x**2+d*x+c)/(b*x**3+a)**(1/2),x)

[Out]

4*3**(1/4)*a**(4/3)*d*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**
(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(-sqrt(3) + 2)*(a**(1/3) + b**(1/3)*x)
*elliptic_e(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3))
+ b**(1/3)*x)), -7 - 4*sqrt(3))/(7*b**(5/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*
x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(a + b*x**3)) - 4*a*e*sqrt(a +
b*x**3)/(9*b**2) - 8*a*d*sqrt(a + b*x**3)/(7*b**(5/3)*(a**(1/3)*(1 + sqrt(3)) +
b**(1/3)*x)) - 4*3**(3/4)*a*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2
)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/
3)*x)*(-10*a**(1/3)*d*(-sqrt(3) + 1) + 7*b**(1/3)*c)*elliptic_f(asin((-a**(1/3)*
(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)), -7 - 4*sqrt
(3))/(105*b**(5/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3))
+ b**(1/3)*x)**2)*sqrt(a + b*x**3)) + 2*c*x*sqrt(a + b*x**3)/(5*b) + 2*d*x**2*s
qrt(a + b*x**3)/(7*b) + 2*e*x**3*sqrt(a + b*x**3)/(9*b)

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Mathematica [C]  time = 2.89881, size = 329, normalized size = 0.56 \[ \frac{-12 i 3^{3/4} a^{4/3} b \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} \left (10 \sqrt [3]{a} d+7 \sqrt [3]{-b} c\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+360 (-1)^{2/3} \sqrt [4]{3} a^{5/3} b d \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+2 (-b)^{2/3} \left (a+b x^3\right ) (b x (63 c+5 x (9 d+7 e x))-70 a e)}{315 (-b)^{8/3} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(c + d*x + e*x^2))/Sqrt[a + b*x^3],x]

[Out]

(2*(-b)^(2/3)*(a + b*x^3)*(-70*a*e + b*x*(63*c + 5*x*(9*d + 7*e*x))) + 360*(-1)^
(2/3)*3^(1/4)*a^(5/3)*b*d*Sqrt[(-1)^(5/6)*(-1 + ((-b)^(1/3)*x)/a^(1/3))]*Sqrt[1
+ ((-b)^(1/3)*x)/a^(1/3) + ((-b)^(2/3)*x^2)/a^(2/3)]*EllipticE[ArcSin[Sqrt[-(-1)
^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)] - (12*I)*3^(3/4)*a^(4/3
)*b*(7*(-b)^(1/3)*c + 10*a^(1/3)*d)*Sqrt[((-1)^(5/6)*(-a^(1/3) + (-b)^(1/3)*x))/
a^(1/3)]*Sqrt[1 + ((-b)^(1/3)*x)/a^(1/3) + ((-b)^(2/3)*x^2)/a^(2/3)]*EllipticF[A
rcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)])/(315*(
-b)^(8/3)*Sqrt[a + b*x^3])

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Maple [A]  time = 0.027, size = 793, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x^2+d*x+c)/(b*x^3+a)^(1/2),x)

[Out]

d*(2/7/b*x^2*(b*x^3+a)^(1/2)+8/21*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))))+c*(2/5/b*x*(b*x^3+a)^(1/2)+4/15*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)*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)-4/9*a/b^2*(b*x^3+a)^(1/
2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^5 + d*x^4 + c*x^3)/sqrt(b*x^3 + a), x)

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Sympy [A]  time = 4.15227, size = 129, normalized size = 0.22 \[ e \left (\begin{cases} - \frac{4 a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases}\right ) + \frac{c x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{7}{3}\right )} + \frac{d x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x**2+d*x+c)/(b*x**3+a)**(1/2),x)

[Out]

e*Piecewise((-4*a*sqrt(a + b*x**3)/(9*b**2) + 2*x**3*sqrt(a + b*x**3)/(9*b), Ne(
b, 0)), (x**6/(6*sqrt(a)), True)) + c*x**4*gamma(4/3)*hyper((1/2, 4/3), (7/3,),
b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(7/3)) + d*x**5*gamma(5/3)*hyper((1/2,
5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(8/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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