Optimal. Leaf size=112 \[ \frac{2 \sqrt{d+e x} (b d-a e)^2}{b^3}+\frac{2 (d+e x)^{3/2} (b d-a e)}{3 b^2}-\frac{2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 (d+e x)^{5/2}}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0585831, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 50, 63, 208} \[ \frac{2 \sqrt{d+e x} (b d-a e)^2}{b^3}+\frac{2 (d+e x)^{3/2} (b d-a e)}{3 b^2}-\frac{2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 (d+e x)^{5/2}}{5 b} \]
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)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^{5/2}}{a+b x} \, dx\\ &=\frac{2 (d+e x)^{5/2}}{5 b}+\frac{(b d-a e) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{b}\\ &=\frac{2 (b d-a e) (d+e x)^{3/2}}{3 b^2}+\frac{2 (d+e x)^{5/2}}{5 b}+\frac{(b d-a e)^2 \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{b^2}\\ &=\frac{2 (b d-a e)^2 \sqrt{d+e x}}{b^3}+\frac{2 (b d-a e) (d+e x)^{3/2}}{3 b^2}+\frac{2 (d+e x)^{5/2}}{5 b}+\frac{(b d-a e)^3 \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{b^3}\\ &=\frac{2 (b d-a e)^2 \sqrt{d+e x}}{b^3}+\frac{2 (b d-a e) (d+e x)^{3/2}}{3 b^2}+\frac{2 (d+e x)^{5/2}}{5 b}+\frac{\left (2 (b d-a e)^3\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^3 e}\\ &=\frac{2 (b d-a e)^2 \sqrt{d+e x}}{b^3}+\frac{2 (b d-a e) (d+e x)^{3/2}}{3 b^2}+\frac{2 (d+e x)^{5/2}}{5 b}-\frac{2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.120746, size = 105, normalized size = 0.94 \[ \frac{2 (b d-a e) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{3 b^{7/2}}+\frac{2 (d+e x)^{5/2}}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 263, normalized size = 2.4 \begin{align*}{\frac{2}{5\,b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ae}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,d}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}{e}^{2}\sqrt{ex+d}}{{b}^{3}}}-4\,{\frac{ade\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{{d}^{2}\sqrt{ex+d}}{b}}-2\,{\frac{{e}^{3}{a}^{3}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+6\,{\frac{d{e}^{2}{a}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-6\,{\frac{a{d}^{2}e}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{{d}^{3}}{\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 = 1.05439, size = 644, normalized size = 5.75 \begin{align*} \left [\frac{15 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\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 (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} +{\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\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 (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} +{\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, b^{3}}\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 = 1.21615, size = 243, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\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^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d + 15 \, \sqrt{x e + d} b^{4} d^{2} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e - 30 \, \sqrt{x e + d} a b^{3} d e + 15 \, \sqrt{x e + d} a^{2} b^{2} e^{2}\right )}}{15 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]