3.396 \(\int \frac{x^3}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=311 \[ -\frac{d x^4 \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt{a+b x^3} \left (a c^2-d^2\right )}+\frac{\sqrt [3]{a c^2-d^2} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 b^{4/3} c^{5/3}}-\frac{\sqrt [3]{a c^2-d^2} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{4/3} c^{5/3}}+\frac{\sqrt [3]{a c^2-d^2} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} c^{5/3}}+\frac{x}{b c} \]

[Out]

x/(b*c) - (d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -(
(b*c^2*x^3)/(a*c^2 - d^2))])/(4*(a*c^2 - d^2)*Sqrt[a + b*x^3]) + ((a*c^2 - d^2)^
(1/3)*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]])/(Sqrt[3]*
b^(4/3)*c^(5/3)) - ((a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3
)*x])/(3*b^(4/3)*c^(5/3)) + ((a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d^2)^(2/3) - b^(1/
3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2])/(6*b^(4/3)*c^(5/3))

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Rubi [A]  time = 1.02733, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345 \[ -\frac{d x^4 \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt{a+b x^3} \left (a c^2-d^2\right )}+\frac{\sqrt [3]{a c^2-d^2} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 b^{4/3} c^{5/3}}-\frac{\sqrt [3]{a c^2-d^2} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{4/3} c^{5/3}}+\frac{\sqrt [3]{a c^2-d^2} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} c^{5/3}}+\frac{x}{b c} \]

Antiderivative was successfully verified.

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

[Out]

x/(b*c) - (d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -(
(b*c^2*x^3)/(a*c^2 - d^2))])/(4*(a*c^2 - d^2)*Sqrt[a + b*x^3]) + ((a*c^2 - d^2)^
(1/3)*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]])/(Sqrt[3]*
b^(4/3)*c^(5/3)) - ((a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3
)*x])/(3*b^(4/3)*c^(5/3)) + ((a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d^2)^(2/3) - b^(1/
3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2])/(6*b^(4/3)*c^(5/3))

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Rubi in Sympy [A]  time = 91.7342, size = 280, normalized size = 0.9 \[ \frac{x}{b c} + \frac{\sqrt [3]{- a c^{2} + d^{2}} \log{\left (\sqrt [3]{b} c^{\frac{2}{3}} x - \sqrt [3]{- a c^{2} + d^{2}} \right )}}{3 b^{\frac{4}{3}} c^{\frac{5}{3}}} - \frac{\sqrt [3]{- a c^{2} + d^{2}} \log{\left (a^{\frac{2}{3}} b^{\frac{2}{3}} c^{\frac{4}{3}} x^{2} + a^{\frac{2}{3}} \sqrt [3]{b} c^{\frac{2}{3}} x \sqrt [3]{- a c^{2} + d^{2}} + a^{\frac{2}{3}} \left (- a c^{2} + d^{2}\right )^{\frac{2}{3}} \right )}}{6 b^{\frac{4}{3}} c^{\frac{5}{3}}} - \frac{\sqrt{3} \sqrt [3]{- a c^{2} + d^{2}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} c^{\frac{2}{3}} x}{3 \sqrt [3]{- a c^{2} + d^{2}}} + \frac{1}{3}\right ) \right )}}{3 b^{\frac{4}{3}} c^{\frac{5}{3}}} + \frac{d x^{4} \sqrt{a + b x^{3}} \operatorname{appellf_{1}}{\left (\frac{4}{3},\frac{1}{2},1,\frac{7}{3},- \frac{b x^{3}}{a},- \frac{b c^{2} x^{3}}{a c^{2} - d^{2}} \right )}}{4 a \sqrt{1 + \frac{b x^{3}}{a}} \left (- a c^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(b*c) + (-a*c**2 + d**2)**(1/3)*log(b**(1/3)*c**(2/3)*x - (-a*c**2 + d**2)**(1
/3))/(3*b**(4/3)*c**(5/3)) - (-a*c**2 + d**2)**(1/3)*log(a**(2/3)*b**(2/3)*c**(4
/3)*x**2 + a**(2/3)*b**(1/3)*c**(2/3)*x*(-a*c**2 + d**2)**(1/3) + a**(2/3)*(-a*c
**2 + d**2)**(2/3))/(6*b**(4/3)*c**(5/3)) - sqrt(3)*(-a*c**2 + d**2)**(1/3)*atan
(sqrt(3)*(2*b**(1/3)*c**(2/3)*x/(3*(-a*c**2 + d**2)**(1/3)) + 1/3))/(3*b**(4/3)*
c**(5/3)) + d*x**4*sqrt(a + b*x**3)*appellf1(4/3, 1/2, 1, 7/3, -b*x**3/a, -b*c**
2*x**3/(a*c**2 - d**2))/(4*a*sqrt(1 + b*x**3/a)*(-a*c**2 + d**2))

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Mathematica [A]  time = 1.4971, size = 470, normalized size = 1.51 \[ \frac{1}{6} \left (\frac{\sqrt [3]{a c^2-d^2} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )-2 \sqrt [3]{a c^2-d^2} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )-2 \sqrt{3} \sqrt [3]{a c^2-d^2} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}-1}{\sqrt{3}}\right )+6 \sqrt [3]{b} c^{2/3} x}{b^{4/3} c^{5/3}}-\frac{21 a d x^4 \left (a c^2-d^2\right ) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{\sqrt{a+b x^3} \left (a c^2+b c^2 x^3-d^2\right ) \left (14 a \left (a c^2-d^2\right ) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )-3 b x^3 \left (2 a c^2 F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )+\left (a c^2-d^2\right ) F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )\right )\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((-21*a*d*(a*c^2 - d^2)*x^4*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^
3)/(a*c^2 - d^2))])/(Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(14*a*(a*c^2 - d^
2)*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 3*b*
x^3*(2*a*c^2*AppellF1[7/3, 1/2, 2, 10/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^
2))] + (a*c^2 - d^2)*AppellF1[7/3, 3/2, 1, 10/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*
c^2 - d^2))]))) + (6*b^(1/3)*c^(2/3)*x - 2*Sqrt[3]*(a*c^2 - d^2)^(1/3)*ArcTan[(-
1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*(a*c^2 - d^2)^(1/3)*
Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + (a*c^2 - d^2)^(1/3)*Log[(a*c^2 -
d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2])/(b^(4
/3)*c^(5/3)))/6

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Maple [C]  time = 0.067, size = 1544, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*I*d/b^2/c^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))+1/3*I/b^4/d*2^(1/2)*sum(1/_alpha
^2*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(
-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*
b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))
/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*
b^2)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*Ellipt
icPi(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),-1/2*c^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*
3^(1/2)*(-a*b^2)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^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)),_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2))*a-1/3*I*d/b^4/c^2*2^(1/2)*sum(1/_
alpha^2*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3
)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)
*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1
/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I
*(-a*b^2)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*E
llipticPi(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),-1/2*c^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2
*b-I*3^(1/2)*(-a*b^2)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/
d^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)),_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2))-1/3*a/c/b^2/(1/c^2/b*(a*c^2-
d^2))^(2/3)*ln(x+(1/c^2/b*(a*c^2-d^2))^(1/3))+1/6*a/c/b^2/(1/c^2/b*(a*c^2-d^2))^
(2/3)*ln(x^2-x*(1/c^2/b*(a*c^2-d^2))^(1/3)+(1/c^2/b*(a*c^2-d^2))^(2/3))-1/3*a/c/
b^2/(1/c^2/b*(a*c^2-d^2))^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c^2/b*(a*c^2-d^
2))^(1/3)*x-1))+x/b/c+1/3/b^2/c^3*d^2/(1/c^2/b*(a*c^2-d^2))^(2/3)*ln(x+(1/c^2/b*
(a*c^2-d^2))^(1/3))-1/6/b^2/c^3*d^2/(1/c^2/b*(a*c^2-d^2))^(2/3)*ln(x^2-x*(1/c^2/
b*(a*c^2-d^2))^(1/3)+(1/c^2/b*(a*c^2-d^2))^(2/3))+1/3/b^2/c^3*d^2/(1/c^2/b*(a*c^
2-d^2))^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c^2/b*(a*c^2-d^2))^(1/3)*x-1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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