Optimal. Leaf size=250 \[ -\frac {\log \left (a^2 b^2 c^{2/3} x^2+2 a b^3 c^{2/3} x+\sqrt [3]{p x^3+q} \left (b^3 \left (-\sqrt [3]{c}\right ) \sqrt [3]{d}-a b^2 \sqrt [3]{c} \sqrt [3]{d} x\right )+b^4 c^{2/3}+b^2 d^{2/3} \left (p x^3+q\right )^{2/3}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log \left (a b \sqrt [3]{c} x+b^2 \sqrt [3]{c}+b \sqrt [3]{d} \sqrt [3]{p x^3+q}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} a \sqrt [3]{c} x+\sqrt {3} b \sqrt [3]{c}}{a \sqrt [3]{c} x+b \sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{p x^3+q}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{d}} \]
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Rubi [F] time = 3.57, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(b+a x) \left (-a q+b p x^2\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(b+a x) \left (-a q+b p x^2\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx &=\int \left (\frac {a b p}{\left (a^3 c+d p\right ) \left (q+p x^3\right )^{2/3}}-\frac {a b \left (b^3 c p+a^3 c q+2 d p q\right )+a^2 \left (3 b^3 c p+a^3 c q+d p q\right ) x+b^2 p \left (2 a^3 c-d p\right ) x^2}{\left (a^3 c+d p\right ) \left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )}\right ) \, dx\\ &=-\frac {\int \frac {a b \left (b^3 c p+a^3 c q+2 d p q\right )+a^2 \left (3 b^3 c p+a^3 c q+d p q\right ) x+b^2 p \left (2 a^3 c-d p\right ) x^2}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx}{a^3 c+d p}+\frac {(a b p) \int \frac {1}{\left (q+p x^3\right )^{2/3}} \, dx}{a^3 c+d p}\\ &=-\frac {\int \left (\frac {a b \left (b^3 c p+a^3 c q+2 d p q\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )}+\frac {a^2 \left (3 b^3 c p+a^3 c q+d p q\right ) x}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )}+\frac {b^2 p \left (2 a^3 c-d p\right ) x^2}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )}\right ) \, dx}{a^3 c+d p}+\frac {\left (a b p \left (1+\frac {p x^3}{q}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {p x^3}{q}\right )^{2/3}} \, dx}{\left (a^3 c+d p\right ) \left (q+p x^3\right )^{2/3}}\\ &=\frac {a b p x \left (1+\frac {p x^3}{q}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {p x^3}{q}\right )}{\left (a^3 c+d p\right ) \left (q+p x^3\right )^{2/3}}-\frac {\left (b^2 p \left (2 a^3 c-d p\right )\right ) \int \frac {x^2}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx}{a^3 c+d p}-\frac {\left (a^2 \left (3 b^3 c p+a^3 c q+d p q\right )\right ) \int \frac {x}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx}{a^3 c+d p}-\frac {\left (a b \left (b^3 c p+a^3 c q+2 d p q\right )\right ) \int \frac {1}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx}{a^3 c+d p}\\ \end {align*}
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Mathematica [F] time = 1.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+a x) \left (-a q+b p x^2\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 17.80, size = 250, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} b \sqrt [3]{c}+\sqrt {3} a \sqrt [3]{c} x}{b \sqrt [3]{c}+a \sqrt [3]{c} x-2 \sqrt [3]{d} \sqrt [3]{q+p x^3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{d}}+\frac {\log \left (b^2 \sqrt [3]{c}+a b \sqrt [3]{c} x+b \sqrt [3]{d} \sqrt [3]{q+p x^3}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\log \left (b^4 c^{2/3}+2 a b^3 c^{2/3} x+a^2 b^2 c^{2/3} x^2+\left (-b^3 \sqrt [3]{c} \sqrt [3]{d}-a b^2 \sqrt [3]{c} \sqrt [3]{d} x\right ) \sqrt [3]{q+p x^3}+b^2 d^{2/3} \left (q+p x^3\right )^{2/3}\right )}{6 c^{2/3} \sqrt [3]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b p x^{2} - a q\right )} {\left (a x + b\right )}}{{\left (3 \, a^{2} b c x^{2} + 3 \, a b^{2} c x + b^{3} c + {\left (a^{3} c + d p\right )} x^{3} + d q\right )} {\left (p x^{3} + q\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +b \right ) \left (b p \,x^{2}-a q \right )}{\left (p \,x^{3}+q \right )^{\frac {2}{3}} \left (b^{3} c +d q +3 a \,b^{2} c x +3 a^{2} b c \,x^{2}+\left (a^{3} c +d p \right ) x^{3}\right )}\, dx\]
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*} \int \frac {{\left (b p x^{2} - a q\right )} {\left (a x + b\right )}}{{\left (3 \, a^{2} b c x^{2} + 3 \, a b^{2} c x + b^{3} c + {\left (a^{3} c + d p\right )} x^{3} + d q\right )} {\left (p x^{3} + q\right )}^{\frac {2}{3}}}\,{d x} \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 {\left (a\,q-b\,p\,x^2\right )\,\left (b+a\,x\right )}{{\left (p\,x^3+q\right )}^{2/3}\,\left (d\,q+x^3\,\left (c\,a^3+d\,p\right )+b^3\,c+3\,a\,b^2\,c\,x+3\,a^2\,b\,c\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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