Optimal. Leaf size=545 \[ -\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}+\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {2 c^{2/5} \left (a+b x^5\right )^{2/5}-\sqrt {5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {2 c^{2/5} \left (a+b x^5\right )^{2/5}+\sqrt {5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \]
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Rubi [A] time = 1.09, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {377, 202, 634, 618, 204, 628, 31} \[ -\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {2 c^{2/5} \left (a+b x^5\right )^{2/5}+2 x^2 (b c-a d)^{2/5}-\sqrt {5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {2 c^{2/5} \left (a+b x^5\right )^{2/5}+2 x^2 (b c-a d)^{2/5}+\sqrt {5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}+\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 202
Rule 204
Rule 377
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^5} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt [5]{c}+\frac {1}{4} \left (1-\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{c^{2/5}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt [5]{c}+\frac {1}{4} \left (1+\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{c^{2/5}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{c}-\sqrt [5]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}\\ &=-\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (5-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{c^{2/5}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{3/5}}+\frac {\left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{c^{2/5}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{3/5}}+\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+2 (b c-a d)^{2/5} x}{c^{2/5}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+2 (b c-a d)^{2/5} x}{c^{2/5}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}\\ &=-\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}-\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}+\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\left (5-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5-\sqrt {5}\right ) c^{2/5} (b c-a d)^{2/5}-x^2} \, dx,x,\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+\frac {2 (b c-a d)^{2/5} x}{\sqrt [5]{a+b x^5}}\right )}{10 c^{3/5}}-\frac {\left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5+\sqrt {5}\right ) c^{2/5} (b c-a d)^{2/5}-x^2} \, dx,x,\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+\frac {2 (b c-a d)^{2/5} x}{\sqrt [5]{a+b x^5}}\right )}{10 c^{3/5}}\\ &=\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c}+\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{c}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {5+\sqrt {5}} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{c}+\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{2 \sqrt {10} \sqrt [5]{c}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}-\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}+\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 48, normalized size = 0.09 \[ \frac {x \, _2F_1\left (\frac {1}{5},1;\frac {6}{5};\frac {(b c-a d) x^5}{c \left (b x^5+a\right )}\right )}{c \sqrt [5]{a+b x^5}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{{\left (b x^{5} + a\right )}^{\frac {1}{5}} {\left (d x^{5} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (d \,x^{5}+c \right )}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{5} + a\right )}^{\frac {1}{5}} {\left (d x^{5} + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^5+a\right )}^{1/5}\,\left (d\,x^5+c\right )} \,d x \]
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 \[ \int \frac {1}{\sqrt [5]{a + b x^{5}} \left (c + d x^{5}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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