Optimal. Leaf size=124 \[ -\frac{2 b \sqrt{d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 b^2 B (d+e x)^{3/2}}{3 e^4} \]
[Out]
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Rubi [A] time = 0.157104, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 b \sqrt{d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 b^2 B (d+e x)^{3/2}}{3 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 53.7933, size = 122, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{4}} + \frac{2 b \sqrt{d + e x} \left (A b e + 2 B a e - 3 B b d\right )}{e^{4}} - \frac{2 \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{e^{4} \sqrt{d + e x}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{2}}{3 e^{4} \left (d + e x\right )^{\frac{3}{2}}} \]
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)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 0.265636, size = 101, normalized size = 0.81 \[ \frac{2 \sqrt{d+e x} \left (-\frac{3 (a e-b d) (a B e+2 A b e-3 b B d)}{d+e x}+\frac{(b d-a e)^2 (B d-A e)}{(d+e x)^2}-b (-6 a B e-3 A b e+8 b B d)+b^2 B e x\right )}{3 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.013, size = 168, normalized size = 1.4 \[ -{\frac{-2\,B{x}^{3}{b}^{2}{e}^{3}-6\,A{b}^{2}{e}^{3}{x}^{2}-12\,Bab{e}^{3}{x}^{2}+12\,B{b}^{2}d{e}^{2}{x}^{2}+12\,Axab{e}^{3}-24\,Ax{b}^{2}d{e}^{2}+6\,Bx{a}^{2}{e}^{3}-48\,Bxabd{e}^{2}+48\,B{b}^{2}{d}^{2}ex+2\,A{a}^{2}{e}^{3}+8\,Aabd{e}^{2}-16\,A{b}^{2}{d}^{2}e+4\,Bd{e}^{2}{a}^{2}-32\,B{d}^{2}abe+32\,B{b}^{2}{d}^{3}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
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)/(e*x+d)^(5/2),x)
[Out]
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Maxima [A] time = 0.74248, size = 220, normalized size = 1.77 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B b^{2} - 3 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284387, size = 221, normalized size = 1.78 \[ \frac{2 \,{\left (B b^{2} e^{3} x^{3} - 16 \, B b^{2} d^{3} - A a^{2} e^{3} + 8 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (2 \, B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} - 3 \,{\left (8 \, B b^{2} d^{2} e - 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )}}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.19404, size = 709, normalized size = 5.72 \[ \begin{cases} - \frac{2 A a^{2} e^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{8 A a b d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 A a b e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 A b^{2} d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 A b^{2} d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 A b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{4 B a^{2} d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 B a^{2} e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{32 B a b d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 B a b d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{12 B a b e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 B b^{2} d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 B b^{2} d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 B b^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 B b^{2} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{A a^{2} x + A a b x^{2} + \frac{A b^{2} x^{3}}{3} + \frac{B a^{2} x^{2}}{2} + \frac{2 B a b x^{3}}{3} + \frac{B b^{2} x^{4}}{4}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
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)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.283127, size = 278, normalized size = 2.24 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{8} - 9 \, \sqrt{x e + d} B b^{2} d e^{8} + 6 \, \sqrt{x e + d} B a b e^{9} + 3 \, \sqrt{x e + d} A b^{2} e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} B b^{2} d^{2} - B b^{2} d^{3} - 12 \,{\left (x e + d\right )} B a b d e - 6 \,{\left (x e + d\right )} A b^{2} d e + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 3 \,{\left (x e + d\right )} B a^{2} e^{2} + 6 \,{\left (x e + d\right )} A a b e^{2} - B a^{2} d e^{2} - 2 \, A a b d e^{2} + A a^{2} e^{3}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]