Optimal. Leaf size=403 \[ \frac {7 \sqrt [6]{b} (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 d^{13/6}}+\frac {7 \sqrt [6]{b} (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}} \]
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Rubi [A] time = 0.53, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {47, 50, 63, 240, 210, 634, 618, 204, 628, 208} \begin {gather*} \frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}+\frac {7 \sqrt [6]{b} (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 d^{13/6}}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}+\frac {7 \sqrt [6]{b} (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} d^{13/6}}-\frac {7 \sqrt [6]{b} (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} d^{13/6}}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 204
Rule 208
Rule 210
Rule 240
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {(7 b) \int \frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}} \, dx}{d}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {(7 b (b c-a d)) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{6 d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {(7 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d x^6}{b}}} \, dx,x,\sqrt [6]{a+b x}\right )}{d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {(7 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}-\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 d^2}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}+\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 d^2}-\frac {\left (7 \sqrt [3]{b} (b c-a d)\right ) \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^2}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}+\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \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 d^{13/6}}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \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 d^{13/6}}-\frac {\left (7 \sqrt [3]{b} (b c-a d)\right ) \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^2}-\frac {\left (7 \sqrt [3]{b} (b c-a d)\right ) \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^2}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}+\frac {7 \sqrt [6]{b} (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 d^{13/6}}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \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 d^{13/6}}+\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \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 d^{13/6}}\\ &=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}+\frac {7 \sqrt [6]{b} (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} d^{13/6}}-\frac {7 \sqrt [6]{b} (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} d^{13/6}}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}+\frac {7 \sqrt [6]{b} (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 d^{13/6}}-\frac {7 \sqrt [6]{b} (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 d^{13/6}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 73, normalized size = 0.18 \begin {gather*} \frac {6 (a+b x)^{13/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac {7}{6},\frac {13}{6};\frac {19}{6};\frac {d (a+b x)}{a d-b c}\right )}{13 b (c+d x)^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 23.49, size = 417, normalized size = 1.03 \begin {gather*} \frac {d^{7/6} (a+b x)^{7/6} \left (-\frac {7 \left (b^{7/6} c-a \sqrt [6]{b} d\right ) \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} d^{13/6}}+\frac {7 \left (b^{7/6} c-a \sqrt [6]{b} d\right ) \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} d^{13/6}}-\frac {7 \left (b^{7/6} c-a \sqrt [6]{b} d\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{a d+b (c+d x)-b c}}\right )}{3 d^{13/6}}-\frac {7 \left (b^{7/6} c-a \sqrt [6]{b} d\right ) \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 d^{13/6}}+\frac {\sqrt [6]{a d+b (c+d x)-b c} (-6 a d+b (c+d x)+6 b c)}{d^{13/6} \sqrt [6]{c+d x}}\right )}{(a d+b d x)^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.43, size = 3084, normalized size = 7.65
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 {7}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x +a \right )^{\frac {7}{6}}}{\left (d x +c \right )^{\frac {7}{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 {7}{6}}}{{\left (d x + c\right )}^{\frac {7}{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 )}^{7/6}}{{\left (c+d\,x\right )}^{7/6}} \,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 {7}{6}}}{\left (c + d x\right )^{\frac {7}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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