Optimal. Leaf size=306 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3 (B d-A e)}{e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)} \]
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Rubi [A] time = 0.468158, antiderivative size = 306, 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 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3 (B d-A e)}{e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 46.8117, size = 304, normalized size = 0.99 \[ \frac{B \left (2 a + 2 b x\right ) \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 b e} + \frac{2 \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (9 A b e - B a e - 8 B b d\right )}{63 b e^{2}} + \frac{4 \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (9 A b e - B a e - 8 B b d\right )}{315 b e^{3}} + \frac{16 \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (9 A b e - B a e - 8 B b d\right )}{315 b e^{4}} + \frac{32 \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (9 A b e - B a e - 8 B b d\right )}{315 b e^{5} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)
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Mathematica [A] time = 0.266441, size = 244, normalized size = 0.8 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (105 a^3 e^3 (3 A e-2 B d+B e x)+63 a^2 b e^2 \left (5 A e (e x-2 d)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-9 a b^2 e \left (3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+b^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{315 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.013, size = 317, normalized size = 1. \[{\frac{70\,B{x}^{4}{b}^{3}{e}^{4}+90\,A{x}^{3}{b}^{3}{e}^{4}+270\,B{x}^{3}a{b}^{2}{e}^{4}-80\,B{x}^{3}{b}^{3}d{e}^{3}+378\,A{x}^{2}a{b}^{2}{e}^{4}-108\,A{x}^{2}{b}^{3}d{e}^{3}+378\,B{x}^{2}{a}^{2}b{e}^{4}-324\,B{x}^{2}a{b}^{2}d{e}^{3}+96\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+630\,Ax{a}^{2}b{e}^{4}-504\,Axa{b}^{2}d{e}^{3}+144\,Ax{b}^{3}{d}^{2}{e}^{2}+210\,Bx{a}^{3}{e}^{4}-504\,Bx{a}^{2}bd{e}^{3}+432\,Bxa{b}^{2}{d}^{2}{e}^{2}-128\,Bx{b}^{3}{d}^{3}e+630\,A{a}^{3}{e}^{4}-1260\,Ad{e}^{3}{a}^{2}b+1008\,Aa{b}^{2}{d}^{2}{e}^{2}-288\,A{b}^{3}{d}^{3}e-420\,Bd{e}^{3}{a}^{3}+1008\,B{a}^{2}b{d}^{2}{e}^{2}-864\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{315\, \left ( bx+a \right ) ^{3}{e}^{5}}\sqrt{ex+d} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.731978, size = 516, normalized size = 1.69 \[ \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 56 \, a b^{2} d^{3} e - 70 \, a^{2} b d^{2} e^{2} + 35 \, a^{3} d e^{3} -{\left (b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} +{\left (2 \, b^{3} d^{2} e^{2} - 7 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} -{\left (8 \, b^{3} d^{3} e - 28 \, a b^{2} d^{2} e^{2} + 35 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} A}{35 \, \sqrt{e x + d} e^{4}} + \frac{2 \,{\left (35 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 432 \, a b^{2} d^{4} e + 504 \, a^{2} b d^{3} e^{2} - 210 \, a^{3} d^{2} e^{3} - 5 \,{\left (b^{3} d e^{4} - 27 \, a b^{2} e^{5}\right )} x^{4} +{\left (8 \, b^{3} d^{2} e^{3} - 27 \, a b^{2} d e^{4} + 189 \, a^{2} b e^{5}\right )} x^{3} -{\left (16 \, b^{3} d^{3} e^{2} - 54 \, a b^{2} d^{2} e^{3} + 63 \, a^{2} b d e^{4} - 105 \, a^{3} e^{5}\right )} x^{2} +{\left (64 \, b^{3} d^{4} e - 216 \, a b^{2} d^{3} e^{2} + 252 \, a^{2} b d^{2} e^{3} - 105 \, a^{3} d e^{4}\right )} x\right )} B}{315 \, \sqrt{e x + d} e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284042, size = 355, normalized size = 1.16 \[ \frac{2 \,{\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \,{\left (8 \, B b^{3} d e^{3} - 9 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{2} e^{2} - 18 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} -{\left (64 \, B b^{3} d^{3} e - 72 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)
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GIAC/XCAS [A] time = 0.292625, size = 582, normalized size = 1.9 \[ \frac{2}{315} \,{\left (105 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{3} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 315 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a^{2} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B a^{2} b e^{\left (-10\right )}{\rm sign}\left (b x + a\right ) + 63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A a b^{2} e^{\left (-10\right )}{\rm sign}\left (b x + a\right ) + 27 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B a b^{2} e^{\left (-21\right )}{\rm sign}\left (b x + a\right ) + 9 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} A b^{3} e^{\left (-21\right )}{\rm sign}\left (b x + a\right ) +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} B b^{3} e^{\left (-36\right )}{\rm sign}\left (b x + a\right ) + 315 \, \sqrt{x e + d} A a^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/sqrt(e*x + d),x, algorithm="giac")
[Out]