Optimal. Leaf size=346 \[ -\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1155 e^4 (a+b x) \sqrt{d+e x} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.188314, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1155 e^4 (a+b x) \sqrt{d+e x} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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)^{11/2}}{\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)^{11/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (11 b^2 e \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (33 e^2 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (231 e^3 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (1155 e^4 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (1155 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{128 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{1155 e^4 (b d-a e) (a+b x) \sqrt{d+e x}}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (1155 e^4 \left (b^2 d-a b e\right )^2 \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^8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{1155 e^4 (b d-a e) (a+b x) \sqrt{d+e x}}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (1155 e^3 \left (b^2 d-a b e\right )^2 \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^8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{1155 e^4 (b d-a e) (a+b x) \sqrt{d+e x}}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1155 e^4 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0491772, size = 67, normalized size = 0.19 \[ -\frac{2 e^4 (a+b x) (d+e x)^{13/2} \, _2F_1\left (5,\frac{13}{2};\frac{15}{2};\frac{b (d+e x)}{b d-a e}\right )}{13 \sqrt{(a+b x)^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.288, size = 1471, normalized size = 4.3 \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{11}{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 [B] time = 1.70008, size = 2079, normalized size = 6.01 \begin{align*} \text{result too large to display} \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 = 1.33739, size = 740, normalized size = 2.14 \begin{align*} \frac{1155 \,{\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\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^{6} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac{2295 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d^{2} e^{4} - 5855 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{4} + 5153 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt{x e + d} b^{5} d^{5} e^{4} - 4590 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} d e^{5} + 17565 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{5} - 20612 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt{x e + d} a b^{4} d^{4} e^{5} + 2295 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{3} e^{6} - 17565 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{6} + 30918 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{6} + 5855 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{7} - 20612 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{7} + 5153 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{8} - 7725 \, \sqrt{x e + d} a^{4} b d e^{8} + 1545 \, \sqrt{x e + d} a^{5} e^{9}}{192 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6} \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{3}{2}} b^{10} e^{4} + 15 \, \sqrt{x e + d} b^{10} d e^{4} - 15 \, \sqrt{x e + d} a b^{9} e^{5}\right )}}{3 \, 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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