Optimal. Leaf size=308 \[ \frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.174307, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {646, 47, 50, 63, 208} \[ \frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (9 e \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 e^2 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{5/2}}{a b+b^2 x} \, dx}{8 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 e^2 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 e^2 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{8 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{63 e^2 (b d-a e)^2 (a+b x) \sqrt{d+e x}}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 e^2 \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{8 b^8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{63 e^2 (b d-a e)^2 (a+b x) \sqrt{d+e x}}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 e \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 b^8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{63 e^2 (b d-a e)^2 (a+b x) \sqrt{d+e x}}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 e^2 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0461776, size = 67, normalized size = 0.22 \[ -\frac{2 e^2 (a+b x) (d+e x)^{11/2} \, _2F_1\left (3,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right )}{11 \sqrt{(a+b x)^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.282, size = 1115, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69455, size = 1535, normalized size = 4.98 \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 = 1.288, size = 602, normalized size = 1.95 \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} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \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} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \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} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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