Optimal. Leaf size=378 \[ -\frac {5 (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{11/6} \sqrt [6]{d}}-\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{11/6} \sqrt [6]{d}}+\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b} \]
________________________________________________________________________________________
Rubi [A] time = 0.48, 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, 240, 210, 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 b^{11/6} \sqrt [6]{d}}+\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 b^{11/6} \sqrt [6]{d}}-\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} b^{11/6} \sqrt [6]{d}}+\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} b^{11/6} \sqrt [6]{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 b^{11/6} \sqrt [6]{d}}+\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 204
Rule 208
Rule 210
Rule 240
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (b c-a d)) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{6 b}\\ &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (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 )}{b^2}\\ &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (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 )}{b^2}\\ &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (b c-a d)) \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 b^{11/6}}+\frac {(5 (b c-a d)) \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 b^{11/6}}+\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 b^{5/3}}\\ &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\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 b^{11/6} \sqrt [6]{d}}+\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 b^{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 b^{5/3}}-\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 b^{11/6} \sqrt [6]{d}}+\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 b^{11/6} \sqrt [6]{d}}\\ &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\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 b^{11/6} \sqrt [6]{d}}-\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 b^{11/6} \sqrt [6]{d}}+\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 b^{11/6} \sqrt [6]{d}}+\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 b^{11/6} \sqrt [6]{d}}-\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 b^{11/6} \sqrt [6]{d}}\\ &=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}-\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} b^{11/6} \sqrt [6]{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} b^{11/6} \sqrt [6]{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 b^{11/6} \sqrt [6]{d}}-\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 b^{11/6} \sqrt [6]{d}}+\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 b^{11/6} \sqrt [6]{d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 71, normalized size = 0.19 \begin {gather*} \frac {6 \sqrt [6]{a+b x} (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},\frac {1}{6};\frac {7}{6};\frac {d (a+b x)}{a d-b c}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 106.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.83, size = 2997, normalized size = 7.93
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {5}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {5}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/6}}{{\left (a+b\,x\right )}^{5/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {5}{6}}}{\left (a + b x\right )^{\frac {5}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________