Optimal. Leaf size=241 \[ \frac{7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac{14 b \sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 d^3}-\frac{7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac{7 \sqrt [3]{b} (b c-a d)^2 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 d^{10/3}}-\frac{14 \sqrt [3]{b} (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} d^{10/3}}-\frac{3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106996, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {47, 50, 59} \[ \frac{7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac{14 b \sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 d^3}-\frac{7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac{7 \sqrt [3]{b} (b c-a d)^2 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 d^{10/3}}-\frac{14 \sqrt [3]{b} (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} d^{10/3}}-\frac{3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{(a+b x)^{7/3}}{(c+d x)^{4/3}} \, dx &=-\frac{3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}+\frac{(7 b) \int \frac{(a+b x)^{4/3}}{\sqrt [3]{c+d x}} \, dx}{d}\\ &=-\frac{3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}+\frac{7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac{(14 b (b c-a d)) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{3 d^2}\\ &=-\frac{3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}-\frac{14 b (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}+\frac{7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}+\frac{\left (14 b (b c-a d)^2\right ) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 d^3}\\ &=-\frac{3 (a+b x)^{7/3}}{d \sqrt [3]{c+d x}}-\frac{14 b (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}+\frac{7 b (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2}-\frac{14 \sqrt [3]{b} (b c-a d)^2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt{3} d^{10/3}}-\frac{7 \sqrt [3]{b} (b c-a d)^2 \log (a+b x)}{9 d^{10/3}}-\frac{7 \sqrt [3]{b} (b c-a d)^2 \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 d^{10/3}}\\ \end{align*}
Mathematica [C] time = 0.0584074, size = 73, normalized size = 0.3 \[ \frac{3 (a+b x)^{10/3} \left (\frac{b (c+d x)}{b c-a d}\right )^{4/3} \, _2F_1\left (\frac{4}{3},\frac{10}{3};\frac{13}{3};\frac{d (a+b x)}{a d-b c}\right )}{10 b (c+d x)^{4/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{7}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{7}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.45291, size = 995, normalized size = 4.13 \begin{align*} -\frac{28 \, \sqrt{3}{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} d \left (-\frac{b}{d}\right )^{\frac{2}{3}} + \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right ) + 14 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 28 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 3 \,{\left (3 \, b^{2} d^{2} x^{2} - 28 \, b^{2} c^{2} + 49 \, a b c d - 18 \, a^{2} d^{2} -{\left (7 \, b^{2} c d - 13 \, a b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{18 \,{\left (d^{4} x + c d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{7}{3}}}{\left (c + d x\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{7}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]