Optimal. Leaf size=251 \[ \frac{e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac{e \sqrt{d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac{(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac{e (d+e x)^{5/2} (2 c d-b e)}{b^2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.534316, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {738, 824, 826, 1166, 208} \[ \frac{e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac{e \sqrt{d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac{(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac{e (d+e x)^{5/2} (2 c d-b e)}{b^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 738
Rule 824
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{\int \frac{(d+e x)^{5/2} \left (\frac{1}{2} d (4 c d-9 b e)-\frac{5}{2} e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{b^2}\\ &=\frac{e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} c d^2 (4 c d-9 b e)-\frac{1}{2} e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac{e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac{e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} c^2 d^3 (4 c d-9 b e)-\frac{1}{2} e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac{e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac{e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} c^3 d^4 (4 c d-9 b e)+\frac{1}{2} e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^3}\\ &=\frac{e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac{e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} c^3 d^4 e (4 c d-9 b e)-\frac{1}{2} d e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right )+\frac{1}{2} e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\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 )}{b^2 c^3}\\ &=\frac{e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac{e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{\left (c d^4 (4 c d-9 b e)\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^3}+\frac{\left ((c d-b e)^4 (4 c d+5 b e)\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^3 c^3}\\ &=\frac{e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac{e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{(c d-b e)^{7/2} (4 c d+5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.395775, size = 202, normalized size = 0.8 \[ \frac{\frac{b \sqrt{d+e x} \left (2 b^2 c^2 e^2 x \left (-9 d^2+13 d e x+e^2 x^2\right )+2 b^3 c e^3 x (19 d-5 e x)-15 b^4 e^4 x-3 b c^3 d^3 (d-4 e x)-6 c^4 d^4 x\right )}{c^3 x (b+c x)}-\frac{3 (c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{7/2}}+3 d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{3 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.28, size = 515, normalized size = 2.1 \begin{align*}{\frac{2\,{e}^{3}}{3\,{c}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{{e}^{4}\sqrt{ex+d}b}{{c}^{3}}}+8\,{\frac{{e}^{3}d\sqrt{ex+d}}{{c}^{2}}}-{\frac{{e}^{5}{b}^{2}}{{c}^{3} \left ( cex+be \right ) }\sqrt{ex+d}}+4\,{\frac{{e}^{4}\sqrt{ex+d}bd}{{c}^{2} \left ( cex+be \right ) }}-6\,{\frac{{e}^{3}\sqrt{ex+d}{d}^{2}}{c \left ( cex+be \right ) }}+4\,{\frac{{e}^{2}\sqrt{ex+d}{d}^{3}}{b \left ( cex+be \right ) }}-{\frac{ce{d}^{4}}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+5\,{\frac{{e}^{5}{b}^{2}}{{c}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-16\,{\frac{{e}^{4}bd}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+14\,{\frac{{e}^{3}{d}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{e}^{2}{d}^{3}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-11\,{\frac{ce{d}^{4}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{5}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{{d}^{4}}{{b}^{2}x}\sqrt{ex+d}}-9\,{\frac{e{d}^{7/2}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{9/2}c}{{b}^{3}}{\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 = 22.8602, size = 3251, normalized size = 12.95 \begin{align*} \text{result too large to display} \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 = 1.43982, size = 589, normalized size = 2.35 \begin{align*} -\frac{{\left (4 \, c d^{5} - 9 \, b d^{4} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (4 \, c^{5} d^{5} - 11 \, b c^{4} d^{4} e + 4 \, b^{2} c^{3} d^{3} e^{2} + 14 \, b^{3} c^{2} d^{2} e^{3} - 16 \, b^{4} c d e^{4} + 5 \, b^{5} e^{5}\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^{3} c^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{4} e^{3} + 12 \, \sqrt{x e + d} c^{4} d e^{3} - 6 \, \sqrt{x e + d} b c^{3} e^{4}\right )}}{3 \, c^{6}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d^{4} e - 2 \, \sqrt{x e + d} c^{4} d^{5} e - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} d^{3} e^{2} + 5 \, \sqrt{x e + d} b c^{3} d^{4} e^{2} + 6 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{2} e^{3} - 6 \, \sqrt{x e + d} b^{2} c^{2} d^{3} e^{3} - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c d e^{4} + 4 \, \sqrt{x e + d} b^{3} c d^{2} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{5} - \sqrt{x e + d} b^{4} d e^{5}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]