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