3.398 \(\int \frac{1}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=300 \[ -\frac{d x \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{\sqrt{a+b x^3} \left (a c^2-d^2\right )}-\frac{\sqrt [3]{c} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}}+\frac{\sqrt [3]{c} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}}-\frac{\sqrt [3]{c} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.608751, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ -\frac{d x \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{\sqrt{a+b x^3} \left (a c^2-d^2\right )}-\frac{\sqrt [3]{c} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}}+\frac{\sqrt [3]{c} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}}-\frac{\sqrt [3]{c} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 83.1627, size = 272, normalized size = 0.91 \[ \frac{\sqrt [3]{c} \log{\left (\sqrt [3]{b} c^{\frac{2}{3}} x - \sqrt [3]{- a c^{2} + d^{2}} \right )}}{3 \sqrt [3]{b} \left (- a c^{2} + d^{2}\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{c} \log{\left (a^{\frac{2}{3}} b^{\frac{2}{3}} c^{\frac{4}{3}} x^{2} + a^{\frac{2}{3}} \sqrt [3]{b} c^{\frac{2}{3}} x \sqrt [3]{- a c^{2} + d^{2}} + a^{\frac{2}{3}} \left (- a c^{2} + d^{2}\right )^{\frac{2}{3}} \right )}}{6 \sqrt [3]{b} \left (- a c^{2} + d^{2}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} \sqrt [3]{c} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} c^{\frac{2}{3}} x}{3 \sqrt [3]{- a c^{2} + d^{2}}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{b} \left (- a c^{2} + d^{2}\right )^{\frac{2}{3}}} + \frac{d x \sqrt{a + b x^{3}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},1,\frac{4}{3},- \frac{b x^{3}}{a},- \frac{b c^{2} x^{3}}{a c^{2} - d^{2}} \right )}}{a \sqrt{1 + \frac{b x^{3}}{a}} \left (- a c^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0502271, size = 0, normalized size = 0. \[ \int \frac{1}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])^(-1),x]

[Out]

Integrate[(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])^(-1), x]

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Maple [C]  time = 0.021, size = 619, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*I/d/b^3*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))*EllipticPi(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/b/c/(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/c/(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/c/(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. \[ \int \frac{1}{b c x^{3} + a c + \sqrt{b x^{3} + a} d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="maxima")

[Out]

integrate(1/(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(1/(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(1/(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. \[ \int \frac{1}{b c x^{3} + a c + \sqrt{b x^{3} + a} d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="giac")

[Out]

integrate(1/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d), x)