Optimal. Leaf size=175 \[ \frac{21 e^2 (d+e x)^{3/2} (b d-a e)}{4 b^4}+\frac{63 e^2 \sqrt{d+e x} (b d-a e)^2}{4 b^5}-\frac{63 e^2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{63 e^2 (d+e x)^{5/2}}{20 b^3} \]
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Rubi [A] time = 0.106475, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {27, 47, 50, 63, 208} \[ \frac{21 e^2 (d+e x)^{3/2} (b d-a e)}{4 b^4}+\frac{63 e^2 \sqrt{d+e x} (b d-a e)^2}{4 b^5}-\frac{63 e^2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{63 e^2 (d+e x)^{5/2}}{20 b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{9/2}}{(a+b x)^3} \, dx\\ &=-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{(9 e) \int \frac{(d+e x)^{7/2}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{\left (63 e^2\right ) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{8 b^2}\\ &=\frac{63 e^2 (d+e x)^{5/2}}{20 b^3}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{\left (63 e^2 (b d-a e)\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{8 b^3}\\ &=\frac{21 e^2 (b d-a e) (d+e x)^{3/2}}{4 b^4}+\frac{63 e^2 (d+e x)^{5/2}}{20 b^3}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{\left (63 e^2 (b d-a e)^2\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{8 b^4}\\ &=\frac{63 e^2 (b d-a e)^2 \sqrt{d+e x}}{4 b^5}+\frac{21 e^2 (b d-a e) (d+e x)^{3/2}}{4 b^4}+\frac{63 e^2 (d+e x)^{5/2}}{20 b^3}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{\left (63 e^2 (b d-a e)^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 b^5}\\ &=\frac{63 e^2 (b d-a e)^2 \sqrt{d+e x}}{4 b^5}+\frac{21 e^2 (b d-a e) (d+e x)^{3/2}}{4 b^4}+\frac{63 e^2 (d+e x)^{5/2}}{20 b^3}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}+\frac{\left (63 e (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 )}{4 b^5}\\ &=\frac{63 e^2 (b d-a e)^2 \sqrt{d+e x}}{4 b^5}+\frac{21 e^2 (b d-a e) (d+e x)^{3/2}}{4 b^4}+\frac{63 e^2 (d+e x)^{5/2}}{20 b^3}-\frac{9 e (d+e x)^{7/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{9/2}}{2 b (a+b x)^2}-\frac{63 e^2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.023888, size = 52, normalized size = 0.3 \[ \frac{2 e^2 (d+e x)^{11/2} \, _2F_1\left (3,\frac{11}{2};\frac{13}{2};-\frac{b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 543, normalized size = 3.1 \begin{align*}{\frac{2\,{e}^{2}}{5\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-2\,{\frac{{e}^{3} \left ( ex+d \right ) ^{3/2}a}{{b}^{4}}}+2\,{\frac{{e}^{2} \left ( ex+d \right ) ^{3/2}d}{{b}^{3}}}+12\,{\frac{{e}^{4}{a}^{2}\sqrt{ex+d}}{{b}^{5}}}-24\,{\frac{{e}^{3}ad\sqrt{ex+d}}{{b}^{4}}}+12\,{\frac{{e}^{2}{d}^{2}\sqrt{ex+d}}{{b}^{3}}}+{\frac{17\,{e}^{5}{a}^{3}}{4\,{b}^{4} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{51\,{e}^{4}{a}^{2}d}{4\,{b}^{3} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{51\,{e}^{3}a{d}^{2}}{4\,{b}^{2} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{17\,{e}^{2}{d}^{3}}{4\,b \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{6}{a}^{4}}{4\,{b}^{5} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-15\,{\frac{{e}^{5}\sqrt{ex+d}d{a}^{3}}{{b}^{4} \left ( bex+ae \right ) ^{2}}}+{\frac{45\,{e}^{4}{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-15\,{\frac{{e}^{3}\sqrt{ex+d}a{d}^{3}}{{b}^{2} \left ( bex+ae \right ) ^{2}}}+{\frac{15\,{e}^{2}{d}^{4}}{4\,b \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{63\,{e}^{5}{a}^{3}}{4\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{189\,{e}^{4}{a}^{2}d}{4\,{b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{189\,{e}^{3}a{d}^{2}}{4\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{63\,{e}^{2}{d}^{3}}{4\,{b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 1.09778, size = 1535, normalized size = 8.77 \begin{align*} \left [\frac{315 \,{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} - 2 \, 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 (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \,{\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \,{\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{40 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{315 \,{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} - 2 \, 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 (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \,{\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \,{\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{20 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.2083, size = 505, normalized size = 2.89 \begin{align*} \frac{63 \,{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{17 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{2} - 15 \, \sqrt{x e + d} b^{4} d^{4} e^{2} - 51 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{3} + 60 \, \sqrt{x e + d} a b^{3} d^{3} e^{3} + 51 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{4} - 90 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{4} - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{5} + 60 \, \sqrt{x e + d} a^{3} b d e^{5} - 15 \, \sqrt{x e + d} a^{4} e^{6}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{5}{2}} b^{12} e^{2} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{12} d e^{2} + 30 \, \sqrt{x e + d} b^{12} d^{2} e^{2} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{11} e^{3} - 60 \, \sqrt{x e + d} a b^{11} d e^{3} + 30 \, \sqrt{x e + d} a^{2} b^{10} e^{4}\right )}}{5 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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