Optimal. Leaf size=404 \[ \frac{x^{13/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{11/2} (5 A b-13 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 x^{9/2} (5 A b-13 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{33 x^{7/2} (5 A b-13 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 x^{5/2} (a+b x) (5 A b-13 a B)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 x^{3/2} (a+b x) (5 A b-13 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a \sqrt{x} (a+b x) (5 A b-13 a B)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{231 a^{3/2} (a+b x) (5 A b-13 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.202163, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {770, 78, 47, 50, 63, 205} \[ \frac{x^{13/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{11/2} (5 A b-13 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 x^{9/2} (5 A b-13 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{33 x^{7/2} (5 A b-13 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 x^{5/2} (a+b x) (5 A b-13 a B)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 x^{3/2} (a+b x) (5 A b-13 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a \sqrt{x} (a+b x) (5 A b-13 a B)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{231 a^{3/2} (a+b x) (5 A b-13 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 78
Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{11/2} (A+B x)}{\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{x^{11/2} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (b^2 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{11/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (11 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (33 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{a b+b^2 x} \, dx}{128 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (231 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{128 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 a (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{128 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a (5 A b-13 a B) \sqrt{x} (a+b x)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (231 a^2 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{128 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a (5 A b-13 a B) \sqrt{x} (a+b x)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (231 a^2 (5 A b-13 a B) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{33 (5 A b-13 a B) x^{7/2}}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{13/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(5 A b-13 a B) x^{11/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (5 A b-13 a B) x^{9/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a (5 A b-13 a B) \sqrt{x} (a+b x)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 (5 A b-13 a B) x^{3/2} (a+b x)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (5 A b-13 a B) x^{5/2} (a+b x)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{231 a^{3/2} (5 A b-13 a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0391307, size = 80, normalized size = 0.2 \[ \frac{x^{13/2} \left (-13 a^4 (a B-A b)-(a+b x)^4 (5 A b-13 a B) \, _2F_1\left (4,\frac{13}{2};\frac{15}{2};-\frac{b x}{a}\right )\right )}{52 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 443, normalized size = 1.1 \begin{align*}{\frac{bx+a}{960\,{b}^{7}} \left ( 384\,B\sqrt{ab}{x}^{13/2}{b}^{6}+640\,A\sqrt{ab}{x}^{11/2}{b}^{6}-1664\,B\sqrt{ab}{x}^{11/2}a{b}^{5}-7040\,A\sqrt{ab}{x}^{9/2}a{b}^{5}+18304\,B\sqrt{ab}{x}^{9/2}{a}^{2}{b}^{4}-46035\,A\sqrt{ab}{x}^{7/2}{a}^{2}{b}^{4}+119691\,B\sqrt{ab}{x}^{7/2}{a}^{3}{b}^{3}-84315\,A\sqrt{ab}{x}^{5/2}{a}^{3}{b}^{3}+17325\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{5}+219219\,B\sqrt{ab}{x}^{5/2}{a}^{4}{b}^{2}-45045\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}{a}^{3}{b}^{4}+69300\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{3}{b}^{4}-180180\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{4}{b}^{3}-63525\,A\sqrt{ab}{x}^{3/2}{a}^{4}{b}^{2}+103950\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{4}{b}^{3}+165165\,B\sqrt{ab}{x}^{3/2}{a}^{5}b-270270\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{5}{b}^{2}+69300\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{5}{b}^{2}-180180\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{6}b-17325\,A\sqrt{ab}\sqrt{x}{a}^{5}b+17325\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{6}b+45045\,B\sqrt{ab}\sqrt{x}{a}^{6}-45045\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{7} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \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 [A] time = 1.41828, size = 1466, normalized size = 3.63 \begin{align*} \left [-\frac{3465 \,{\left (13 \, B a^{6} - 5 \, A a^{5} b +{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \,{\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \,{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x}}{1920 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}, -\frac{3465 \,{\left (13 \, B a^{6} - 5 \, A a^{5} b +{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \,{\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \,{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x}}{960 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\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 [A] time = 1.19864, size = 294, normalized size = 0.73 \begin{align*} -\frac{231 \,{\left (13 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{7} \mathrm{sgn}\left (b x + a\right )} + \frac{4431 \, B a^{3} b^{3} x^{\frac{7}{2}} - 2295 \, A a^{2} b^{4} x^{\frac{7}{2}} + 11767 \, B a^{4} b^{2} x^{\frac{5}{2}} - 5855 \, A a^{3} b^{3} x^{\frac{5}{2}} + 10633 \, B a^{5} b x^{\frac{3}{2}} - 5153 \, A a^{4} b^{2} x^{\frac{3}{2}} + 3249 \, B a^{6} \sqrt{x} - 1545 \, A a^{5} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{7} \mathrm{sgn}\left (b x + a\right )} + \frac{2 \,{\left (3 \, B b^{20} x^{\frac{5}{2}} - 25 \, B a b^{19} x^{\frac{3}{2}} + 5 \, A b^{20} x^{\frac{3}{2}} + 225 \, B a^{2} b^{18} \sqrt{x} - 75 \, A a b^{19} \sqrt{x}\right )}}{15 \, b^{25} \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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