Optimal. Leaf size=64 \[ \frac {x^4 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};1,\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a \sqrt {c+d x^3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {525, 524}
\begin {gather*} \frac {x^4 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {4}{3};1,\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {x^3}{\left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {c+d x^3}}\\ &=\frac {x^4 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};1,\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [A]
time = 10.04, size = 65, normalized size = 1.02 \begin {gather*} \frac {x^4 \sqrt {\frac {c+d x^3}{c}} F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )}{4 a \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
6.
time = 0.34, size = 719, normalized size = 11.23 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^{3}}{\left (a + b x^{3}\right ) \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^3}{\left (b\,x^3+a\right )\,\sqrt {d\,x^3+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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