Optimal. Leaf size=67 \[ \frac {x^2 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {2}{3};2,\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 c \sqrt {c+d x^3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {525, 524}
\begin {gather*} \frac {x^2 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {2}{3};2,\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 c \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}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {x}{\left (a+b x^3\right )^2 \left (1+\frac {d x^3}{c}\right )^{3/2}} \, dx}{c \sqrt {c+d x^3}}\\ &=\frac {x^2 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {2}{3};2,\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 c \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(67)=134\).
time = 10.18, size = 216, normalized size = 3.22 \begin {gather*} -\frac {x^2 \left (-10 a \left (2 a^2 d^2+2 a b d^2 x^3+b^2 c \left (c+d x^3\right )\right )+5 \left (-b^2 c^2+6 a b c d+a^2 d^2\right ) \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+b d (b c+2 a d) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}{30 a^2 c (b c-a d)^2 \left (a+b x^3\right ) \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 = 986, normalized size = 14.72
method | result | size |
default | \(\text {Expression too large to display}\) | \(986\) |
elliptic | \(\text {Expression too large to display}\) | \(986\) |
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}{\left (a + b x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, 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 {x}{{\left (b\,x^3+a\right )}^2\,{\left (d\,x^3+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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