Optimal. Leaf size=118 \[ \frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e (d+e x)^{3/2}}{3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.227992, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {703, 824, 826, 1166, 208} \[ \frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e (d+e x)^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 703
Rule 824
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{b x+c x^2} \, dx &=\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{\int \frac{\sqrt{d+e x} \left (c d^2+e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{c}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{\int \frac{c^2 d^3+e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{c^2}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{2 \operatorname{Subst}\left (\int \frac{c^2 d^3 e-d e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )+e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{c^2}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{\left (2 c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}-\frac{\left (2 (c d-b e)^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b c^2}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.123229, size = 107, normalized size = 0.91 \[ \frac{2 e \sqrt{d+e x} (-3 b e+7 c d+c e x)}{3 c^2}+\frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.223, size = 237, normalized size = 2. \begin{align*}{\frac{2\,e}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{\sqrt{ex+d}b{e}^{2}}{{c}^{2}}}+4\,{\frac{de\sqrt{ex+d}}{c}}+2\,{\frac{{e}^{3}{b}^{2}}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-6\,{\frac{bd{e}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+6\,{\frac{e{d}^{2}}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{c{d}^{3}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{d}^{5/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \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 = 3.86695, size = 1350, normalized size = 11.44 \begin{align*} \left [\frac{3 \, c^{2} d^{\frac{5}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 3 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right ) + 2 \,{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{3 \, c^{2} d^{\frac{5}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 6 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (-\frac{\sqrt{e x + d} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right ) + 2 \,{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{6 \, c^{2} \sqrt{-d} d^{2} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) + 3 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right ) + 2 \,{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{2 \,{\left (3 \, c^{2} \sqrt{-d} d^{2} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) + 3 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (-\frac{\sqrt{e x + d} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right ) +{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}\right )}}{3 \, b c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 56.1243, size = 119, normalized size = 1.01 \begin{align*} \frac{2 e \left (d + e x\right )^{\frac{3}{2}}}{3 c} + \frac{\sqrt{d + e x} \left (- 2 b e^{2} + 4 c d e\right )}{c^{2}} + \frac{2 d^{3} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b \sqrt{- d}} + \frac{2 \left (b e - c d\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{b c^{3} \sqrt{\frac{b e - c d}{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.34448, size = 217, normalized size = 1.84 \begin{align*} \frac{2 \, d^{3} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} - \frac{2 \,{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e + 6 \, \sqrt{x e + d} c^{2} d e - 3 \, \sqrt{x e + d} b c e^{2}\right )}}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]