Optimal. Leaf size=100 \[ \frac{8 x}{15 a^6 c^3 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0199559, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {40, 39} \[ \frac{8 x}{15 a^6 c^3 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx &=\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{4 \int \frac{1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{5 a^2 c}\\ &=\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{8 \int \frac{1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{15 a^4 c^2}\\ &=\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{8 x}{15 a^6 c^3 \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}
Mathematica [A] time = 0.0302893, size = 57, normalized size = 0.57 \[ \frac{-20 a^2 b^2 x^3+15 a^4 x+8 b^4 x^5}{15 a^6 c (a+b x)^{5/2} (c (a-b x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 56, normalized size = 0.6 \begin{align*}{\frac{ \left ( -bx+a \right ) x \left ( 8\,{b}^{4}{x}^{4}-20\,{x}^{2}{a}^{2}{b}^{2}+15\,{a}^{4} \right ) }{15\,{a}^{6}} \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.996784, size = 107, normalized size = 1.07 \begin{align*} \frac{x}{5 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{5}{2}} a^{2} c} + \frac{4 \, x}{15 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{3}{2}} a^{4} c^{2}} + \frac{8 \, x}{15 \, \sqrt{-b^{2} c x^{2} + a^{2} c} a^{6} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75966, size = 203, normalized size = 2.03 \begin{align*} -\frac{{\left (8 \, b^{4} x^{5} - 20 \, a^{2} b^{2} x^{3} + 15 \, a^{4} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{15 \,{\left (a^{6} b^{6} c^{4} x^{6} - 3 \, a^{8} b^{4} c^{4} x^{4} + 3 \, a^{10} b^{2} c^{4} x^{2} - a^{12} c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [B] time = 1.42209, size = 494, normalized size = 4.94 \begin{align*} -\frac{\sqrt{-b c x + a c}{\left ({\left (b c x - a c\right )}{\left (\frac{275 \, c}{a^{5} b{\left | c \right |}} + \frac{64 \,{\left (b c x - a c\right )}}{a^{6} b{\left | c \right |}}\right )} + \frac{300 \, c^{2}}{a^{4} b{\left | c \right |}}\right )}}{240 \,{\left (2 \, a c^{2} +{\left (b c x - a c\right )} c\right )}^{\frac{5}{2}}} - \frac{1024 \, a^{4} c^{8} - 2200 \, a^{3}{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2} c^{6} + 1660 \, a^{2}{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{4} c^{4} - 450 \, a{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{6} c^{2} + 45 \,{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{8}}{60 \,{\left (2 \, a c^{2} -{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}\right )}^{5} a^{5} b \sqrt{-c}{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]