Optimal. Leaf size=304 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^5 (a+b x)} \]
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Rubi [A] time = 0.142961, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^{5/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^{5/2}}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{3/2}}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 \sqrt{d+e x}}+\frac{b^5 (-4 b B d+A b e+3 a B e) \sqrt{d+e x}}{e^4}+\frac{b^6 B (d+e x)^{3/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^3 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^{3/2}}+\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt{d+e x}}+\frac{6 b (b d-a e) (2 b B d-A b e-a B e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}+\frac{2 b^3 B (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.189254, size = 241, normalized size = 0.79 \[ \frac{2 \sqrt{(a+b x)^2} \left (15 a^2 b e^2 \left (B \left (8 d^2+12 d e x+3 e^2 x^2\right )-A e (2 d+3 e x)\right )-5 a^3 e^3 (A e+2 B d+3 B e x)+15 a b^2 e \left (A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )\right )+b^3 \left (5 A e \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )+B \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )\right )}{15 e^5 (a+b x) (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 317, normalized size = 1. \begin{align*} -{\frac{-6\,B{x}^{4}{b}^{3}{e}^{4}-10\,A{x}^{3}{b}^{3}{e}^{4}-30\,B{x}^{3}a{b}^{2}{e}^{4}+16\,B{x}^{3}{b}^{3}d{e}^{3}-90\,A{x}^{2}a{b}^{2}{e}^{4}+60\,A{x}^{2}{b}^{3}d{e}^{3}-90\,B{x}^{2}{a}^{2}b{e}^{4}+180\,B{x}^{2}a{b}^{2}d{e}^{3}-96\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+90\,Ax{a}^{2}b{e}^{4}-360\,Axa{b}^{2}d{e}^{3}+240\,Ax{b}^{3}{d}^{2}{e}^{2}+30\,Bx{a}^{3}{e}^{4}-360\,Bx{a}^{2}bd{e}^{3}+720\,Bxa{b}^{2}{d}^{2}{e}^{2}-384\,Bx{b}^{3}{d}^{3}e+10\,A{a}^{3}{e}^{4}+60\,Ad{e}^{3}{a}^{2}b-240\,Aa{b}^{2}{d}^{2}{e}^{2}+160\,A{b}^{3}{d}^{3}e+20\,Bd{e}^{3}{a}^{3}-240\,B{a}^{2}b{d}^{2}{e}^{2}+480\,Ba{b}^{2}{d}^{3}e-256\,B{b}^{3}{d}^{4}}{15\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09398, size = 410, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} A}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (3 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} - 240 \, a b^{2} d^{3} e + 120 \, a^{2} b d^{2} e^{2} - 10 \, a^{3} d e^{3} -{\left (8 \, b^{3} d e^{3} - 15 \, a b^{2} e^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{2} e^{2} - 30 \, a b^{2} d e^{3} + 15 \, a^{2} b e^{4}\right )} x^{2} + 3 \,{\left (64 \, b^{3} d^{3} e - 120 \, a b^{2} d^{2} e^{2} + 60 \, a^{2} b d e^{3} - 5 \, a^{3} e^{4}\right )} x\right )} B}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35636, size = 605, normalized size = 1.99 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} - 5 \, A a^{3} e^{4} - 80 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 120 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (8 \, B b^{3} d e^{3} - 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{2} e^{2} - 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (64 \, B b^{3} d^{3} e - 40 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 60 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\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 = 1.21495, size = 687, normalized size = 2.26 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} e^{20} \mathrm{sgn}\left (b x + a\right ) - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d e^{20} \mathrm{sgn}\left (b x + a\right ) + 90 \, \sqrt{x e + d} B b^{3} d^{2} e^{20} \mathrm{sgn}\left (b x + a\right ) + 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} e^{21} \mathrm{sgn}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} e^{21} \mathrm{sgn}\left (b x + a\right ) - 135 \, \sqrt{x e + d} B a b^{2} d e^{21} \mathrm{sgn}\left (b x + a\right ) - 45 \, \sqrt{x e + d} A b^{3} d e^{21} \mathrm{sgn}\left (b x + a\right ) + 45 \, \sqrt{x e + d} B a^{2} b e^{22} \mathrm{sgn}\left (b x + a\right ) + 45 \, \sqrt{x e + d} A a b^{2} e^{22} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} B b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) - B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) - 27 \,{\left (x e + d\right )} B a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) - 9 \,{\left (x e + d\right )} A b^{3} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) + A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 18 \,{\left (x e + d\right )} B a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) + 18 \,{\left (x e + d\right )} A a b^{2} d e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \,{\left (x e + d\right )} B a^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 9 \,{\left (x e + d\right )} A a^{2} b e^{3} \mathrm{sgn}\left (b x + a\right ) + B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) - A a^{3} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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