Optimal. Leaf size=378 \[ \frac {5 (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{11/6}}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d} \]
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Rubi [A] time = 0.56, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {50, 63, 331, 296, 634, 618, 204, 628, 208} \begin {gather*} \frac {5 (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{11/6}}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 204
Rule 208
Rule 296
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/6}}{(c+d x)^{5/6}} \, dx &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{6 d}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {x^4}{\left (c-\frac {a d}{b}+\frac {d x^6}{b}\right )^{5/6}} \, dx,x,\sqrt [6]{a+b x}\right )}{b d}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {x^4}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b d}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{3 d^{5/3}}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}-\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{5/3}}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}+\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{5/3}}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{11/6}}+\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {-\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}+\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{4 d^{5/3}}+\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{4 d^{5/3}}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}+\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{11/6}}-\frac {(5 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{11/6}}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 73, normalized size = 0.19 \begin {gather*} \frac {6 (a+b x)^{11/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {11}{6};\frac {17}{6};\frac {d (a+b x)}{a d-b c}\right )}{11 b (c+d x)^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 17.90, size = 385, normalized size = 1.02 \begin {gather*} \frac {d^{5/6} (a+b x)^{5/6} \left (\frac {\sqrt [6]{c+d x} (a d+b (c+d x)-b c)^{5/6}}{d^{11/6}}+\frac {5 (b c-a d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{b} \sqrt [6]{c+d x}-2 \sqrt [6]{a d+b (c+d x)-b c}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{a d+b (c+d x)-b c}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{a d+b (c+d x)-b c}}\right )}{3 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [3]{a d+b (c+d x)-b c}+\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [6]{b} \sqrt [6]{c+d x} \sqrt [6]{a d+b (c+d x)-b c}}\right )}{6 \sqrt [6]{b} d^{11/6}}\right )}{(a d+b d x)^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.25, size = 2997, normalized size = 7.93
result too large to display
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 x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {5}{6}}}\,{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 \begin {gather*} \int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (d x +c \right )^{\frac {5}{6}}}\, dx \end {gather*}
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 x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {5}{6}}}\,{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+b\,x\right )}^{5/6}}{{\left (c+d\,x\right )}^{5/6}} \,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*} \int \frac {\left (a + b x\right )^{\frac {5}{6}}}{\left (c + d x\right )^{\frac {5}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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