Optimal. Leaf size=449 \[ \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{19/6}} \]
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Rubi [A]
time = 0.45, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {49, 52, 65,
338, 302, 648, 632, 210, 642, 214} \begin {gather*} \frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}+\frac {91 d (a+b x)^{5/6} \sqrt [6]{c+d x} (b c-a d)}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 210
Rule 214
Rule 302
Rule 338
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx &=-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {(13 d) \int \frac {(c+d x)^{7/6}}{\sqrt [6]{a+b x}} \, dx}{b}\\ &=\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {(91 d (b c-a d)) \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx}{12 b^2}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 d (b c-a d)^2\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{72 b^3}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 d (b c-a d)^2\right ) \text {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 )}{12 b^4}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 d (b c-a d)^2\right ) \text {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 )}{12 b^4}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 \sqrt [3]{d} (b c-a d)^2\right ) \text {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 )}{36 b^{19/6}}+\frac {\left (91 \sqrt [3]{d} (b c-a d)^2\right ) \text {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 )}{36 b^{19/6}}+\frac {\left (91 \sqrt [3]{d} (b c-a d)^2\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 )}{36 b^3}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}-\frac {\left (91 \sqrt [6]{d} (b c-a d)^2\right ) \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 )}{144 b^{19/6}}+\frac {\left (91 \sqrt [6]{d} (b c-a d)^2\right ) \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 )}{144 b^{19/6}}-\frac {\left (91 \sqrt [3]{d} (b c-a d)^2\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 )}{48 b^3}-\frac {\left (91 \sqrt [3]{d} (b c-a d)^2\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 )}{48 b^3}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{144 b^{19/6}}-\frac {\left (91 \sqrt [6]{d} (b c-a d)^2\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 )}{24 b^{19/6}}+\frac {\left (91 \sqrt [6]{d} (b c-a d)^2\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 )}{24 b^{19/6}}\\ &=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{24 \sqrt {3} b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \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 )}{144 b^{19/6}}\\ \end {align*}
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Mathematica [A]
time = 0.92, size = 308, normalized size = 0.69 \begin {gather*} \frac {(b c-a d)^2 \left (\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x} \left (-91 a^2 d^2-13 a b d (-13 c+d x)+b^2 \left (-72 c^2+25 c d x+6 d^2 x^2\right )\right )}{(b c-a d)^2 \sqrt [6]{a+b x}}-91 \sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}}{\sqrt {3}}\right )+91 \sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}}{\sqrt {3}}\right )+182 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+91 \sqrt [6]{d} \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 )\right )}{72 b^{19/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {13}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5690 vs.
\(2 (339) = 678\).
time = 1.35, size = 5690, normalized size = 12.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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*} \text {could not integrate} \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 (c+d\,x\right )}^{13/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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