Optimal. Leaf size=124 \[ \frac{2 b (-2 a B e-A b e+3 b B d)}{e^4 \sqrt{d+e x}}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4 (d+e x)^{3/2}}+\frac{2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 b^2 B \sqrt{d+e x}}{e^4} \]
[Out]
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Rubi [A] time = 0.155115, 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 \[ \frac{2 b (-2 a B e-A b e+3 b B d)}{e^4 \sqrt{d+e x}}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4 (d+e x)^{3/2}}+\frac{2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 b^2 B \sqrt{d+e x}}{e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 53.9399, size = 122, normalized size = 0.98 \[ \frac{2 B b^{2} \sqrt{d + e x}}{e^{4}} - \frac{2 b \left (A b e + 2 B a e - 3 B b d\right )}{e^{4} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{3 e^{4} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{2}}{5 e^{4} \left (d + e x\right )^{\frac{5}{2}}} \]
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)**(7/2),x)
[Out]
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Mathematica [A] time = 0.304035, size = 107, normalized size = 0.86 \[ \frac{2 \sqrt{d+e x} \left (-\frac{15 b (2 a B e+A b e-3 b B d)}{d+e x}-\frac{5 (a e-b d) (a B e+2 A b e-3 b B d)}{(d+e x)^2}+\frac{3 (b d-a e)^2 (B d-A e)}{(d+e x)^3}+15 b^2 B\right )}{15 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.013, size = 169, normalized size = 1.4 \[ -{\frac{-30\,B{x}^{3}{b}^{2}{e}^{3}+30\,A{b}^{2}{e}^{3}{x}^{2}+60\,Bab{e}^{3}{x}^{2}-180\,B{b}^{2}d{e}^{2}{x}^{2}+20\,Axab{e}^{3}+40\,Ax{b}^{2}d{e}^{2}+10\,Bx{a}^{2}{e}^{3}+80\,Bxabd{e}^{2}-240\,B{b}^{2}{d}^{2}ex+6\,A{a}^{2}{e}^{3}+8\,Aabd{e}^{2}+16\,A{b}^{2}{d}^{2}e+4\,Bd{e}^{2}{a}^{2}+32\,B{d}^{2}abe-96\,B{b}^{2}{d}^{3}}{15\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
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)^(7/2),x)
[Out]
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Maxima [A] time = 0.728231, size = 221, normalized size = 1.78 \[ \frac{2 \,{\left (\frac{15 \, \sqrt{e x + d} B b^{2}}{e^{3}} + \frac{3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} - 3 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{3}}\right )}}{15 \, e} \]
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)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285156, size = 239, normalized size = 1.93 \[ \frac{2 \,{\left (15 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} - 8 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (6 \, B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \,{\left (24 \, B b^{2} d^{2} e - 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} -{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )}}{15 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} \]
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)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.6656, size = 1015, normalized size = 8.19 \[ \text{result too large to display} \]
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)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.277874, size = 273, normalized size = 2.2 \[ 2 \, \sqrt{x e + d} B b^{2} e^{\left (-4\right )} + \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} B b^{2} d - 15 \,{\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} - 30 \,{\left (x e + d\right )}^{2} B a b e - 15 \,{\left (x e + d\right )}^{2} A b^{2} e + 20 \,{\left (x e + d\right )} B a b d e + 10 \,{\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \,{\left (x e + d\right )} B a^{2} e^{2} - 10 \,{\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]
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)^(7/2),x, algorithm="giac")
[Out]