Optimal. Leaf size=95 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}-\frac{35 a \sqrt{x}}{4 b^4}-\frac{x^{7/2}}{2 b (a+b x)^2}+\frac{35 x^{3/2}}{12 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0326997, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 205} \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}-\frac{35 a \sqrt{x}}{4 b^4}-\frac{x^{7/2}}{2 b (a+b x)^2}+\frac{35 x^{3/2}}{12 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{(a+b x)^3} \, dx &=-\frac{x^{7/2}}{2 b (a+b x)^2}+\frac{7 \int \frac{x^{5/2}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac{x^{7/2}}{2 b (a+b x)^2}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}+\frac{35 \int \frac{x^{3/2}}{a+b x} \, dx}{8 b^2}\\ &=\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a+b x)^2}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}-\frac{(35 a) \int \frac{\sqrt{x}}{a+b x} \, dx}{8 b^3}\\ &=-\frac{35 a \sqrt{x}}{4 b^4}+\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a+b x)^2}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}+\frac{\left (35 a^2\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{8 b^4}\\ &=-\frac{35 a \sqrt{x}}{4 b^4}+\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a+b x)^2}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}+\frac{\left (35 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{4 b^4}\\ &=-\frac{35 a \sqrt{x}}{4 b^4}+\frac{35 x^{3/2}}{12 b^3}-\frac{x^{7/2}}{2 b (a+b x)^2}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}+\frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0041032, size = 27, normalized size = 0.28 \[ \frac{2 x^{9/2} \, _2F_1\left (3,\frac{9}{2};\frac{11}{2};-\frac{b x}{a}\right )}{9 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 79, normalized size = 0.8 \begin{align*}{\frac{2}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-6\,{\frac{a\sqrt{x}}{{b}^{4}}}-{\frac{13\,{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{11\,{a}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,{a}^{2}}{4\,{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.32586, size = 509, normalized size = 5.36 \begin{align*} \left [\frac{105 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt{x}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{105 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt{x}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\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 [A] time = 1.25841, size = 104, normalized size = 1.09 \begin{align*} \frac{35 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{4}} - \frac{13 \, a^{2} b x^{\frac{3}{2}} + 11 \, a^{3} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{4}} + \frac{2 \,{\left (b^{6} x^{\frac{3}{2}} - 9 \, a b^{5} \sqrt{x}\right )}}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]