Optimal. Leaf size=88 \[ \frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 b d}-\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 b^{3/2} d^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {446, 80, 63, 217, 206} \[ \frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 b d}-\frac {(a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 b^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 206
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 b d}-\frac {(b c+a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^3\right )}{6 b d}\\ &=\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 b d}-\frac {(b c+a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b^2 d}\\ &=\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 b d}-\frac {(b c+a d) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^3}}{\sqrt {c+d x^3}}\right )}{3 b^2 d}\\ &=\frac {\sqrt {a+b x^3} \sqrt {c+d x^3}}{3 b d}-\frac {(b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 b^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 123, normalized size = 1.40 \[ \frac {b \sqrt {d} \sqrt {a+b x^3} \left (c+d x^3\right )-\sqrt {b c-a d} (a d+b c) \sqrt {\frac {b \left (c+d x^3\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b c-a d}}\right )}{3 b^2 d^{3/2} \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 256, normalized size = 2.91 \[ \left [\frac {4 \, \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} b d + {\left (b c + a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{6} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{3} - 4 \, {\left (2 \, b d x^{3} + b c + a d\right )} \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} \sqrt {b d}\right )}{12 \, b^{2} d^{2}}, \frac {2 \, \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} b d + {\left (b c + a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{6} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{3}\right )}}\right )}{6 \, b^{2} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 104, normalized size = 1.18 \[ \frac {\frac {{\left (b c + a d\right )} \log \left ({\left | -\sqrt {b x^{3} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d} + \frac {\sqrt {b x^{3} + a} \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d}}{b d}}{3 \, {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.25, size = 283, normalized size = 3.22 \[ \frac {\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,\left (\frac {2\,a\,d}{3}+\frac {2\,b\,c}{3}\right )}{d^3\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}+\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\frac {2\,a\,d}{3}+\frac {2\,b\,c}{3}\right )}{b\,d^2\,{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3}-\frac {8\,\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^2}{3\,d^2\,{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^4}{{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^4}+\frac {b^2}{d^2}-\frac {2\,b\,{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^2}{d\,{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}\right )\,\left (a\,d+b\,c\right )}{3\,b^{3/2}\,d^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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