Optimal. Leaf size=162 \[ \frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.182752, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \[ \frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^{9/2}}{(a+b x)^2} \, dx\\ &=-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{(9 e) \int \frac{(d+e x)^{7/2}}{a+b x} \, dx}{2 b}\\ &=\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{(9 e (b d-a e)) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 e (b d-a e)^2\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 e (b d-a e)^3\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{2 b^4}\\ &=\frac{9 e (b d-a e)^3 \sqrt{d+e x}}{b^5}+\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 e (b d-a e)^4\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 b^5}\\ &=\frac{9 e (b d-a e)^3 \sqrt{d+e x}}{b^5}+\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{\left (9 (b d-a e)^4\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^5}\\ &=\frac{9 e (b d-a e)^3 \sqrt{d+e x}}{b^5}+\frac{3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac{9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac{9 e (d+e x)^{7/2}}{7 b^2}-\frac{(d+e x)^{9/2}}{b (a+b x)}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.024523, size = 50, normalized size = 0.31 \[ \frac{2 e (d+e x)^{11/2} \, _2F_1\left (2,\frac{11}{2};\frac{13}{2};-\frac{b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.213, size = 539, normalized size = 3.3 \begin{align*}{\frac{2\,e}{7\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a{e}^{2}}{5\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{4\,de}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+2\,{\frac{ \left ( ex+d \right ) ^{3/2}{a}^{2}{e}^{3}}{{b}^{4}}}-4\,{\frac{ \left ( ex+d \right ) ^{3/2}ad{e}^{2}}{{b}^{3}}}+2\,{\frac{e \left ( ex+d \right ) ^{3/2}{d}^{2}}{{b}^{2}}}-8\,{\frac{{a}^{3}{e}^{4}\sqrt{ex+d}}{{b}^{5}}}+24\,{\frac{{a}^{2}d{e}^{3}\sqrt{ex+d}}{{b}^{4}}}-24\,{\frac{a{d}^{2}{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+8\,{\frac{e{d}^{3}\sqrt{ex+d}}{{b}^{2}}}-{\frac{{a}^{4}{e}^{5}}{{b}^{5} \left ( bxe+ae \right ) }\sqrt{ex+d}}+4\,{\frac{\sqrt{ex+d}{a}^{3}d{e}^{4}}{{b}^{4} \left ( bxe+ae \right ) }}-6\,{\frac{\sqrt{ex+d}{d}^{2}{e}^{3}{a}^{2}}{{b}^{3} \left ( bxe+ae \right ) }}+4\,{\frac{\sqrt{ex+d}a{d}^{3}{e}^{2}}{{b}^{2} \left ( bxe+ae \right ) }}-{\frac{e{d}^{4}}{b \left ( bxe+ae \right ) }\sqrt{ex+d}}+9\,{\frac{{a}^{4}{e}^{5}}{{b}^{5}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-36\,{\frac{{a}^{3}d{e}^{4}}{{b}^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+54\,{\frac{{d}^{2}{e}^{3}{a}^{2}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-36\,{\frac{a{d}^{3}{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+9\,{\frac{e{d}^{4}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\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 [B] time = 2.0296, size = 1450, normalized size = 8.95 \begin{align*} \left [-\frac{315 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\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 (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \,{\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{70 \,{\left (b^{6} x + a b^{5}\right )}}, -\frac{315 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\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 (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \,{\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (b^{6} x + a b^{5}\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 [B] time = 1.26353, size = 522, normalized size = 3.22 \begin{align*} \frac{9 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\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^{5}} - \frac{\sqrt{x e + d} b^{4} d^{4} e - 4 \, \sqrt{x e + d} a b^{3} d^{3} e^{2} + 6 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{3} - 4 \, \sqrt{x e + d} a^{3} b d e^{4} + \sqrt{x e + d} a^{4} e^{5}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{12} e + 14 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{12} d e + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{12} d^{2} e + 140 \, \sqrt{x e + d} b^{12} d^{3} e - 14 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{11} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{11} d e^{2} - 420 \, \sqrt{x e + d} a b^{11} d^{2} e^{2} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{10} e^{3} + 420 \, \sqrt{x e + d} a^{2} b^{10} d e^{3} - 140 \, \sqrt{x e + d} a^{3} b^{9} e^{4}\right )}}{35 \, b^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]