Optimal. Leaf size=319 \[ -\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}} \]
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Rubi [A]
time = 0.28, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2187, 331,
298, 31, 648, 631, 210, 642, 525, 524} \begin {gather*} \frac {d \sqrt {\frac {b x^3}{a}+1} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}+\frac {\sqrt [3]{b} c^{5/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {c}{x \left (a c^2-d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 524
Rule 525
Rule 631
Rule 642
Rule 648
Rule 2187
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx &=(a c) \int \frac {1}{x^2 \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx-(a d) \int \frac {1}{x^2 \sqrt {a+b x^3} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx\\ &=-\frac {c}{\left (a c^2-d^2\right ) x}-\frac {\left (a b c^3\right ) \int \frac {x}{a^2 c^2-a d^2+a b c^2 x^3} \, dx}{a c^2-d^2}-\frac {\left (a d \sqrt {1+\frac {b x^3}{a}}\right ) \int \frac {1}{x^2 \sqrt {1+\frac {b x^3}{a}} \left (a^2 c^2-a d^2+a b c^2 x^3\right )} \, dx}{\sqrt {a+b x^3}}\\ &=-\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\left (\sqrt [3]{a} b^{2/3} c^{7/3}\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x} \, dx}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (\sqrt [3]{a} b^{2/3} c^{7/3}\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{a c^2-d^2}+\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{3 \left (a c^2-d^2\right )^{4/3}}\\ &=-\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (\sqrt [3]{b} c^{5/3}\right ) \int \frac {-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}+2 a^{2/3} b^{2/3} c^{4/3} x}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{6 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (a^{2/3} b^{2/3} c^{7/3}\right ) \int \frac {1}{a^{2/3} \left (a c^2-d^2\right )^{2/3}-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} b^{2/3} c^{4/3} x^2} \, dx}{2 \left (a c^2-d^2\right )}\\ &=-\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}}-\frac {\left (\sqrt [3]{b} c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{\left (a c^2-d^2\right )^{4/3}}\\ &=-\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} F_1\left (-\frac {1}{3};\frac {1}{2},1;\frac {2}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 10.44, size = 496, normalized size = 1.55 \begin {gather*} \frac {15 b d \sqrt [3]{a c^2-d^2} \left (a c^2+d^2\right ) x^3 \sqrt {1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-6 b^2 c^2 d \sqrt [3]{a c^2-d^2} x^6 \sqrt {1+\frac {b x^3}{a}} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-10 \left (a c^2-d^2\right ) \left (-6 a d \sqrt [3]{a c^2-d^2}-6 b d \sqrt [3]{a c^2-d^2} x^3+6 a c \sqrt [3]{a c^2-d^2} \sqrt {a+b x^3}+2 \sqrt {3} a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \tan ^{-1}\left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )-2 a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )+a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )\right )}{60 a \left (a c^2-d^2\right )^{7/3} x \sqrt {a+b 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 = 0.14, size = 2227, normalized size = 6.98
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1202\) |
default | \(\text {Expression too large to display}\) | \(2227\) |
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 {1}{x^{2} \left (a c + b c x^{3} + d \sqrt {a + b x^{3}}\right )}\, 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 = -0.00 \begin {gather*} \int \frac {1}{x^2\,\left (a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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