Optimal. Leaf size=124 \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
[Out]
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Rubi [A] time = 0.160414, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 29.2582, size = 122, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{3}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{e^{4}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{2}}{e^{4} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(B*x+A)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.169023, size = 135, normalized size = 1.09 \[ \frac{30 a^2 e^2 (-A e+2 B d+B e x)+20 a b e \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+2 b^2 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{15 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 169, normalized size = 1.4 \[ -{\frac{-6\,B{b}^{2}{x}^{3}{e}^{3}-10\,A{b}^{2}{e}^{3}{x}^{2}-20\,Bab{e}^{3}{x}^{2}+12\,B{b}^{2}d{e}^{2}{x}^{2}-60\,Aab{e}^{3}x+40\,A{b}^{2}d{e}^{2}x-30\,B{a}^{2}{e}^{3}x+80\,Babd{e}^{2}x-48\,B{b}^{2}{d}^{2}ex+30\,{a}^{2}A{e}^{3}-120\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-60\,B{a}^{2}d{e}^{2}+160\,Bab{d}^{2}e-96\,B{b}^{2}{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 1.36962, size = 225, normalized size = 1.81 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b^{2} - 5 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}}{\sqrt{e x + d} e^{3}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224437, size = 209, normalized size = 1.69 \[ \frac{2 \,{\left (3 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 15 \, A a^{2} e^{3} - 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 30 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} -{\left (6 \, B b^{2} d e^{2} - 5 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 20 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )}}{15 \, \sqrt{e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{2}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(B*x+A)/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214049, size = 296, normalized size = 2.39 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{2} e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{16} + 45 \, \sqrt{x e + d} B b^{2} d^{2} e^{16} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{17} - 60 \, \sqrt{x e + d} B a b d e^{17} - 30 \, \sqrt{x e + d} A b^{2} d e^{17} + 15 \, \sqrt{x e + d} B a^{2} e^{18} + 30 \, \sqrt{x e + d} A a b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2/(e*x + d)^(3/2),x, algorithm="giac")
[Out]