Optimal. Leaf size=177 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5+b}}\right )}{c^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5+b}}\right )}{c^{3/4}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b}}\right )}{\sqrt {2} c^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^5+b}}{\sqrt {a x^5+b}+\sqrt {c} x^2}\right )}{\sqrt {2} c^{3/4}} \]
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Rubi [F] time = 4.26, 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 {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {c x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx &=c \int \frac {x^6 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx\\ &=c \int \left (\frac {x}{a \left (b+a x^5\right )^{3/4}}+\frac {x \left (-b^2-6 a b x^5+c^2 x^8\right )}{a \left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )}\right ) \, dx\\ &=\frac {c \int \frac {x}{\left (b+a x^5\right )^{3/4}} \, dx}{a}+\frac {c \int \frac {x \left (-b^2-6 a b x^5+c^2 x^8\right )}{\left (b+a x^5\right )^{3/4} \left (b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right )} \, dx}{a}\\ &=\frac {c \int \left (\frac {-b c^2-a b c x-5 a^2 b x^2+c^3 x^4}{2 a c \left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )}+\frac {b c^2-a b c x+5 a^2 b x^2+c^3 x^4}{2 a c \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}\right ) \, dx}{a}+\frac {\left (c \left (1+\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x}{\left (1+\frac {a x^5}{b}\right )^{3/4}} \, dx}{a \left (b+a x^5\right )^{3/4}}\\ &=\frac {c x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};-\frac {a x^5}{b}\right )}{2 a \left (b+a x^5\right )^{3/4}}+\frac {\int \frac {-b c^2-a b c x-5 a^2 b x^2+c^3 x^4}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )} \, dx}{2 a^2}+\frac {\int \frac {b c^2-a b c x+5 a^2 b x^2+c^3 x^4}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{2 a^2}\\ &=\frac {c x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};-\frac {a x^5}{b}\right )}{2 a \left (b+a x^5\right )^{3/4}}+\frac {\int \left (-\frac {c^3 x^4}{\left (-b+c x^4-a x^5\right ) \left (b+a x^5\right )^{3/4}}-\frac {b c^2}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )}-\frac {a b c x}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )}-\frac {5 a^2 b x^2}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )}\right ) \, dx}{2 a^2}+\frac {\int \left (\frac {b c^2}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}-\frac {a b c x}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}+\frac {5 a^2 b x^2}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}+\frac {c^3 x^4}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )}\right ) \, dx}{2 a^2}\\ &=\frac {c x^2 \left (1+\frac {a x^5}{b}\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};-\frac {a x^5}{b}\right )}{2 a \left (b+a x^5\right )^{3/4}}-\frac {1}{2} (5 b) \int \frac {x^2}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )} \, dx+\frac {1}{2} (5 b) \int \frac {x^2}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx-\frac {(b c) \int \frac {x}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )} \, dx}{2 a}-\frac {(b c) \int \frac {x}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{2 a}-\frac {\left (b c^2\right ) \int \frac {1}{\left (b+a x^5\right )^{3/4} \left (b-c x^4+a x^5\right )} \, dx}{2 a^2}+\frac {\left (b c^2\right ) \int \frac {1}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{2 a^2}-\frac {c^3 \int \frac {x^4}{\left (-b+c x^4-a x^5\right ) \left (b+a x^5\right )^{3/4}} \, dx}{2 a^2}+\frac {c^3 \int \frac {x^4}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx}{2 a^2}\\ \end {align*}
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Mathematica [F] time = 1.21, size = 56, normalized size = 0.32 \begin {gather*} c \int \frac {x^6 \left (a x^5-4 b\right )}{\left (a x^5+b\right )^{3/4} \left (a^2 x^{10}+2 a b x^5+b^2-c^2 x^8\right )} \, dx \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 14.27, size = 176, normalized size = 0.99 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )}{c^{3/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {b+a x^5}}\right )}{\sqrt {2} c^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b+a x^5}}\right )}{c^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{\sqrt {2} c^{3/4}} \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 (a x^{5} - 4 \, b\right )} c x^{6}}{{\left (a^{2} x^{10} - c^{2} x^{8} + 2 \, a b x^{5} + b^{2}\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{6} \left (a \,x^{5}-4 b \right )}{\left (a \,x^{5}+b \right )^{\frac {3}{4}} \left (a^{2} x^{10}-c^{2} x^{8}+2 a b \,x^{5}+b^{2}\right )}\, dx\]
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*} c \int \frac {{\left (a x^{5} - 4 \, b\right )} x^{6}}{{\left (a^{2} x^{10} - c^{2} x^{8} + 2 \, a b x^{5} + b^{2}\right )} {\left (a x^{5} + b\right )}^{\frac {3}{4}}}\,{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 = -0.01 \begin {gather*} -\int \frac {c\,x^6\,\left (4\,b-a\,x^5\right )}{{\left (a\,x^5+b\right )}^{3/4}\,\left (a^2\,x^{10}+2\,a\,b\,x^5+b^2-c^2\,x^8\right )} \,d x \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*} c \left (\int \frac {a x^{11}}{a^{2} x^{10} \left (a x^{5} + b\right )^{\frac {3}{4}} + 2 a b x^{5} \left (a x^{5} + b\right )^{\frac {3}{4}} + b^{2} \left (a x^{5} + b\right )^{\frac {3}{4}} - c^{2} x^{8} \left (a x^{5} + b\right )^{\frac {3}{4}}}\, dx + \int \left (- \frac {4 b x^{6}}{a^{2} x^{10} \left (a x^{5} + b\right )^{\frac {3}{4}} + 2 a b x^{5} \left (a x^{5} + b\right )^{\frac {3}{4}} + b^{2} \left (a x^{5} + b\right )^{\frac {3}{4}} - c^{2} x^{8} \left (a x^{5} + b\right )^{\frac {3}{4}}}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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