Optimal. Leaf size=48 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 \sqrt {b} \sqrt {d}} \]
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Rubi [A] time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {444, 63, 217, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 \sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 217
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {2 \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}\\ &=\frac {2 \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}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 \sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 85, normalized size = 1.77 \begin {gather*} \frac {2 \sqrt {c+d x^3} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b c-a d}}\right )}{3 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^3\right )}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.85, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {d} \sqrt {a+b x^3}}\right )}{3 \sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 194, normalized size = 4.04 \begin {gather*} \left [\frac {\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 )}{6 \, b d}, -\frac {\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 )}{3 \, b d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 54, normalized size = 1.12 \begin {gather*} -\frac {2 \, b \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 )}{3 \, \sqrt {b d} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.04, size = 49, normalized size = 1.02 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}{\sqrt {-b\,d}\,\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}\right )}{3\,\sqrt {-b\,d}} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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