Optimal. Leaf size=332 \[ -\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}-\frac {\sqrt {3} \sqrt [6]{b} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}+\frac {\sqrt {3} \sqrt [6]{b} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}+\frac {2 \sqrt [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{7/6}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.33, antiderivative size = 332, 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 = {49, 65, 246,
216, 648, 632, 210, 642, 214} \begin {gather*} -\frac {\sqrt {3} \sqrt [6]{b} \text {ArcTan}\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 )}{d^{7/6}}+\frac {\sqrt {3} \sqrt [6]{b} \text {ArcTan}\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 )}{d^{7/6}}-\frac {\sqrt [6]{b} \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 )}{2 d^{7/6}}+\frac {\sqrt [6]{b} \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 )}{2 d^{7/6}}+\frac {2 \sqrt [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 65
Rule 210
Rule 214
Rule 216
Rule 246
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}+\frac {b \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{d}\\ &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}+\frac {6 \text {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}\\ &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}+\frac {6 \text {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}\\ &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}+\frac {\left (2 \sqrt [6]{b}\right ) \text {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 )}{d}+\frac {\left (2 \sqrt [6]{b}\right ) \text {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 )}{d}+\frac {\left (2 \sqrt [3]{b}\right ) \text {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 )}{d}\\ &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}+\frac {2 \sqrt [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}-\frac {\sqrt [6]{b} \text {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 )}{2 d^{7/6}}+\frac {\sqrt [6]{b} \text {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 )}{2 d^{7/6}}+\frac {\left (3 \sqrt [3]{b}\right ) \text {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 )}{2 d}+\frac {\left (3 \sqrt [3]{b}\right ) \text {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 )}{2 d}\\ &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}+\frac {2 \sqrt [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}-\frac {\sqrt [6]{b} \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 )}{2 d^{7/6}}+\frac {\sqrt [6]{b} \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 )}{2 d^{7/6}}+\frac {\left (3 \sqrt [6]{b}\right ) \text {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 )}{d^{7/6}}-\frac {\left (3 \sqrt [6]{b}\right ) \text {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 )}{d^{7/6}}\\ &=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}-\frac {\sqrt {3} \sqrt [6]{b} \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 )}{d^{7/6}}+\frac {\sqrt {3} \sqrt [6]{b} \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 )}{d^{7/6}}+\frac {2 \sqrt [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}-\frac {\sqrt [6]{b} \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 )}{2 d^{7/6}}+\frac {\sqrt [6]{b} \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 )}{2 d^{7/6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.30, size = 239, normalized size = 0.72 \begin {gather*} \frac {-\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}-\sqrt {3} \sqrt [6]{b} \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 )+\sqrt {3} \sqrt [6]{b} \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 [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+\sqrt [6]{b} \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{d^{7/6}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {1}{6}}}{\left (d x +c \right )^{\frac {7}{6}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 663 vs.
\(2 (239) = 478\).
time = 1.49, size = 663, normalized size = 2.00 \begin {gather*} -\frac {4 \, \sqrt {3} {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} d^{6} \left (\frac {b}{d^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (d^{7} x + c d^{6}\right )} \sqrt {\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} d \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} + {\left (d^{3} x + c d^{2}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}} \left (\frac {b}{d^{7}}\right )^{\frac {5}{6}} + \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) + 4 \, \sqrt {3} {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} d^{6} \left (\frac {b}{d^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (d^{7} x + c d^{6}\right )} \sqrt {-\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} d \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} - {\left (d^{3} x + c d^{2}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}} \left (\frac {b}{d^{7}}\right )^{\frac {5}{6}} - \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) - {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} d \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} + {\left (d^{3} x + c d^{2}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )}}{d x + c}\right ) + {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} d \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} - {\left (d^{3} x + c d^{2}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )}}{d x + c}\right ) - 2 \, {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) + 2 \, {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) + 12 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{2 \, {\left (d^{2} x + c d\right )}} \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 {\sqrt [6]{a + b x}}{\left (c + d x\right )^{\frac {7}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (a+b\,x\right )}^{1/6}}{{\left (c+d\,x\right )}^{7/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________