Optimal. Leaf size=86 \[ \frac{2 \sqrt{d+e x} (b d-a e)}{b^2}-\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 (d+e x)^{3/2}}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0431648, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 50, 63, 208} \[ \frac{2 \sqrt{d+e x} (b d-a e)}{b^2}-\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 (d+e x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^{3/2}}{a+b x} \, dx\\ &=\frac{2 (d+e x)^{3/2}}{3 b}+\frac{(b d-a e) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{b}\\ &=\frac{2 (b d-a e) \sqrt{d+e x}}{b^2}+\frac{2 (d+e x)^{3/2}}{3 b}+\frac{(b d-a e)^2 \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{b^2}\\ &=\frac{2 (b d-a e) \sqrt{d+e x}}{b^2}+\frac{2 (d+e x)^{3/2}}{3 b}+\frac{\left (2 (b d-a e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b^2 e}\\ &=\frac{2 (b d-a e) \sqrt{d+e x}}{b^2}+\frac{2 (d+e x)^{3/2}}{3 b}-\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0580442, size = 77, normalized size = 0.9 \[ \frac{2 \sqrt{d+e x} (-3 a e+4 b d+b e x)}{3 b^2}-\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.008, size = 167, normalized size = 1.9 \begin{align*}{\frac{2}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{ae\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{d\sqrt{ex+d}}{b}}+2\,{\frac{{a}^{2}{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-4\,{\frac{ade}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{{d}^{2}}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \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 = 1.12656, size = 424, normalized size = 4.93 \begin{align*} \left [-\frac{3 \,{\left (b d - a e\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}}{3 \, b^{2}}, -\frac{2 \,{\left (3 \,{\left (b d - a e\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (-\frac{\sqrt{e x + d} b \sqrt{-\frac{b d - a e}{b}}}{b d - a e}\right ) -{\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}\right )}}{3 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 106.509, size = 82, normalized size = 0.95 \begin{align*} \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 b} + \frac{\sqrt{d + e x} \left (- 2 a e + 2 b d\right )}{b^{2}} + \frac{2 \left (a e - b d\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b^{3} \sqrt{\frac{a e - b d}{b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20168, size = 151, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} + 3 \, \sqrt{x e + d} b^{2} d - 3 \, \sqrt{x e + d} a b e\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]