Optimal. Leaf size=280 \[ -\frac{(d+e x)^{5/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{3 e (a+b x) \sqrt{d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (a+b x) (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]
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Rubi [A] time = 0.229144, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {770, 78, 47, 50, 63, 208} \[ -\frac{(d+e x)^{5/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{3 e (a+b x) \sqrt{d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (a+b x) (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 78
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^{3/2}}{\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{(A+B x) (d+e x)^{3/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((4 b B d+A b e-5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 e (4 b B d+A b e-5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{8 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{3 e (4 b B d+A b e-5 a B e) (a+b x) \sqrt{d+e x}}{4 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 e \left (b^2 d-a b e\right ) (4 b B d+A b e-5 a B e) \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^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{3 e (4 b B d+A b e-5 a B e) (a+b x) \sqrt{d+e x}}{4 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 \left (b^2 d-a b e\right ) (4 b B d+A b e-5 a B e) \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^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{3 e (4 b B d+A b e-5 a B e) (a+b x) \sqrt{d+e x}}{4 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+A b e-5 a B e) (d+e x)^{3/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{5/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (4 b B d+A b e-5 a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{b d-a e} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0961638, size = 110, normalized size = 0.39 \[ \frac{(a+b x) (d+e x)^{5/2} \left (\frac{e (a+b x)^2 (-5 a B e+A b e+4 b B d) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+5 a B-5 A b\right )}{10 b \left ((a+b x)^2\right )^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 608, normalized size = 2.2 \begin{align*} -{\frac{bx+a}{4\,{b}^{3}e} \left ( -3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}{b}^{3}{e}^{3}+15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}a{b}^{2}{e}^{3}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}{b}^{3}d{e}^{2}-6\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) xa{b}^{2}{e}^{3}-8\,B\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{x}^{2}{b}^{2}{e}^{2}+30\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) x{a}^{2}b{e}^{3}-24\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) xa{b}^{2}d{e}^{2}+5\,A\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{b}^{2}e-3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{2}b{e}^{3}-9\,B\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}abe+4\,B\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{b}^{2}d-16\,B\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}xab{e}^{2}+15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{3}{e}^{3}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{2}bd{e}^{2}+3\,A\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}ab{e}^{2}-3\,A\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{2}de-15\,B\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{2}{e}^{2}+11\,B\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}abde-4\,B\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \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 (B x + A\right )}{\left (e x + d\right )}^{\frac{3}{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 = 1.66612, size = 1432, normalized size = 5.11 \begin{align*} \left [\frac{3 \,{\left (4 \, B a^{2} b d e -{\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (2 \,{\left (B a b^{3} + A b^{4}\right )} d^{2} -{\left (17 \, B a^{2} b^{2} - A a b^{3}\right )} d e + 3 \,{\left (5 \, B a^{3} b - A a^{2} b^{2}\right )} e^{2} - 8 \,{\left (B b^{4} d e - B a b^{3} e^{2}\right )} x^{2} +{\left (4 \, B b^{4} d^{2} -{\left (29 \, B a b^{3} - 5 \, A b^{4}\right )} d e + 5 \,{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (a^{2} b^{5} d - a^{3} b^{4} e +{\left (b^{7} d - a b^{6} e\right )} x^{2} + 2 \,{\left (a b^{6} d - a^{2} b^{5} e\right )} x\right )}}, \frac{3 \,{\left (4 \, B a^{2} b d e -{\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (2 \,{\left (B a b^{3} + A b^{4}\right )} d^{2} -{\left (17 \, B a^{2} b^{2} - A a b^{3}\right )} d e + 3 \,{\left (5 \, B a^{3} b - A a^{2} b^{2}\right )} e^{2} - 8 \,{\left (B b^{4} d e - B a b^{3} e^{2}\right )} x^{2} +{\left (4 \, B b^{4} d^{2} -{\left (29 \, B a b^{3} - 5 \, A b^{4}\right )} d e + 5 \,{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (a^{2} b^{5} d - a^{3} b^{4} e +{\left (b^{7} d - a b^{6} e\right )} x^{2} + 2 \,{\left (a b^{6} d - a^{2} b^{5} e\right )} x\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 = 1.22162, size = 421, normalized size = 1.5 \begin{align*} \frac{2 \, \sqrt{x e + d} B e}{b^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac{3 \,{\left (4 \, B b d e^{2} - 5 \, B a e^{3} + A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt{-b^{2} d + a b e} b^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac{{\left (4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{2} - 4 \, \sqrt{x e + d} B b^{2} d^{2} e^{2} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{3} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{3} + 11 \, \sqrt{x e + d} B a b d e^{3} - 3 \, \sqrt{x e + d} A b^{2} d e^{3} - 7 \, \sqrt{x e + d} B a^{2} e^{4} + 3 \, \sqrt{x e + d} A a b e^{4}\right )} e^{\left (-1\right )}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3} \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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