Optimal. Leaf size=66 \[ -\frac{16 d \sqrt [4]{a+b x}}{3 \sqrt [4]{c+d x} (b c-a d)^2}-\frac{4}{3 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.008442, 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} \[ -\frac{16 d \sqrt [4]{a+b x}}{3 \sqrt [4]{c+d x} (b c-a d)^2}-\frac{4}{3 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{7/4} (c+d x)^{5/4}} \, dx &=-\frac{4}{3 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac{(4 d) \int \frac{1}{(a+b x)^{3/4} (c+d x)^{5/4}} \, dx}{3 (b c-a d)}\\ &=-\frac{4}{3 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}-\frac{16 d \sqrt [4]{a+b x}}{3 (b c-a d)^2 \sqrt [4]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0170147, size = 45, normalized size = 0.68 \[ -\frac{4 (3 a d+b (c+4 d x))}{3 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 53, normalized size = 0.8 \begin{align*} -{\frac{16\,bdx+12\,ad+4\,bc}{3\,{a}^{2}{d}^{2}-6\,abcd+3\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{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{7}{4}}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.68122, size = 270, normalized size = 4.09 \begin{align*} -\frac{4 \,{\left (4 \, b d x + b c + 3 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{3 \,{\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{7}{4}} \left (c + d x\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]