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

Optimal. Leaf size=65 \[ \frac{c x^2 \sqrt{c+d x^3} F_1\left (\frac{2}{3};1,-\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a \sqrt{\frac{d x^3}{c}+1}} \]

[Out]

(c*x^2*Sqrt[c + d*x^3]*AppellF1[2/3, 1, -3/2, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/
(2*a*Sqrt[1 + (d*x^3)/c])

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Rubi [A]  time = 0.140085, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c x^2 \sqrt{c+d x^3} F_1\left (\frac{2}{3};1,-\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a \sqrt{\frac{d x^3}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2*Sqrt[c + d*x^3]*AppellF1[2/3, 1, -3/2, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/
(2*a*Sqrt[1 + (d*x^3)/c])

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Rubi in Sympy [A]  time = 20.3618, size = 53, normalized size = 0.82 \[ \frac{c x^{2} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{3}{2},1,\frac{5}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**2*sqrt(c + d*x**3)*appellf1(2/3, -3/2, 1, 5/3, -d*x**3/c, -b*x**3/a)/(2*a*s
qrt(1 + d*x**3/c))

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Mathematica [B]  time = 0.698085, size = 437, normalized size = 6.72 \[ \frac{x^2 \left (\frac{25 a c^2 (4 a d-7 b c) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{2 d \left (15 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c \left (10 a c+3 a d x^3+20 b c x^3+10 b d x^6\right ) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}{3 x^3 \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}\right )}{35 b \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^2*((25*a*c^2*(-7*b*c + 4*a*d)*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x
^3)/a)])/(-10*a*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3
*(2*b*c*AppellF1[5/3, 1/2, 2, 8/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[5/
3, 3/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])) + (2*d*(-8*a*c*(10*a*c + 20*b*c*x^
3 + 3*a*d*x^3 + 10*b*d*x^6)*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a
)] + 15*x^3*(a + b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[8/3, 1/2, 2, 11/3, -((d*x^3)
/c), -((b*x^3)/a)] + a*d*AppellF1[8/3, 3/2, 1, 11/3, -((d*x^3)/c), -((b*x^3)/a)]
)))/(-16*a*c*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b
*c*AppellF1[8/3, 1/2, 2, 11/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[8/3, 3
/2, 1, 11/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(35*b*(a + b*x^3)*Sqrt[c + d*x^3])

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Maple [C]  time = 0.05, size = 930, normalized size = 14.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7/b*d*x^2*(d*x^3+c)^(1/2)-2/3*I*(-d*(a*d-2*b*c)/b^2-4/7/b*d*c)*3^(1/2)/d*(-c*d
^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(
-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2
)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^
(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1
/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-
1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c
*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(
-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(
-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2
/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+1/3*I/b^2/d^2*2^(1/2)
*sum((-a^2*d^2+2*a*b*c*d-b^2*c^2)/_alpha/(a*d-b*c)*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+
1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*
(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2
*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c
)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/
3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-
c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2
*b/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^
(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(
-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3
*b+a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(c + d*x**3)**(3/2)/(a + b*x**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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