Optimal. Leaf size=66 \[ -\frac{9 d \sqrt [3]{a+b x}}{2 \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0086929, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{9 d \sqrt [3]{a+b x}}{2 \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{5/3} (c+d x)^{4/3}} \, dx &=-\frac{3}{2 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac{(3 d) \int \frac{1}{(a+b x)^{2/3} (c+d x)^{4/3}} \, dx}{2 (b c-a d)}\\ &=-\frac{3}{2 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac{9 d \sqrt [3]{a+b x}}{2 (b c-a d)^2 \sqrt [3]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0161732, size = 45, normalized size = 0.68 \[ -\frac{3 (2 a d+b (c+3 d x))}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 53, normalized size = 0.8 \begin{align*} -{\frac{9\,bdx+6\,ad+3\,bc}{2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}} \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{1}{{\left (b x + a\right )}^{\frac{5}{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 = 2.31814, size = 270, normalized size = 4.09 \begin{align*} -\frac{3 \,{\left (3 \, b d x + b c + 2 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\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{1}{\left (a + b x\right )^{\frac{5}{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{1}{{\left (b x + a\right )}^{\frac{5}{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]