Optimal. Leaf size=294 \[ -\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 e^4 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.14133, antiderivative size = 294, 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} \[ -\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 e^4 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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 )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (9 b^2 e \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (21 e^2 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (315 e^4 \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{128 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (315 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{128 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (315 e^3 \left (b^2 d-a b e\right ) \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 )}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 e^4 \sqrt{b d-a e} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0450814, size = 67, normalized size = 0.23 \[ -\frac{2 e^4 (a+b x) (d+e x)^{11/2} \, _2F_1\left (5,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right )}{11 \sqrt{(a+b x)^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.282, size = 892, normalized size = 3. \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67546, size = 1453, normalized size = 4.94 \begin{align*} \left [\frac{315 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\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 (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} -{\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{128 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac{315 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\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 (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} -{\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{64 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} 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 = 1.31808, size = 560, normalized size = 1.9 \begin{align*} \frac{315 \,{\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \, \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{2 \, \sqrt{x e + d} e^{4}}{b^{5} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac{325 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{4} - 765 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{4} + 643 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt{x e + d} b^{4} d^{4} e^{4} - 325 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{5} + 1530 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{5} - 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt{x e + d} a b^{3} d^{3} e^{5} - 765 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{6} + 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{6} - 643 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{7} + 748 \, \sqrt{x e + d} a^{3} b d e^{7} - 187 \, \sqrt{x e + d} a^{4} e^{8}}{64 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5} \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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