Optimal. Leaf size=88 \[ -\frac {\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {525, 524}
\begin {gather*} -\frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \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 {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {b x^3}{a}} \int \frac {1}{x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3}} \, dx}{\sqrt {a+b x^3}}\\ &=\frac {\left (\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}\right ) \int \frac {1}{x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\\ &=-\frac {\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(88)=176\).
time = 2.49, size = 365, normalized size = 4.15 \begin {gather*} \frac {b d x^6 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 \left (-4 a c \left (2 a c+3 b c x^3+3 a d x^3+2 b d x^6\right ) F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (a d F_1\left (\frac {4}{3};\frac {1}{2},\frac {3}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c F_1\left (\frac {4}{3};\frac {3}{2},\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{8 a c F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d F_1\left (\frac {4}{3};\frac {1}{2},\frac {3}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c F_1\left (\frac {4}{3};\frac {3}{2},\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}}{8 a c x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \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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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 = -0.01 \begin {gather*} \int \frac {1}{x^3\,\sqrt {b\,x^3+a}\,\sqrt {d\,x^3+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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