Optimal. Leaf size=112 \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 c \sqrt{d+e x} (3 B d-A e)}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]
[Out]
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Rubi [A] time = 0.136142, antiderivative size = 112, 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 \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 c \sqrt{d+e x} (3 B d-A e)}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2))/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 23.9895, size = 110, normalized size = 0.98 \[ \frac{2 B c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{4}} + \frac{2 c \sqrt{d + e x} \left (A e - 3 B d\right )}{e^{4}} - \frac{2 \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{e^{4} \sqrt{d + e x}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{3 e^{4} \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 0.115748, size = 94, normalized size = 0.84 \[ -\frac{2 \left (a A e^3+a B e^2 (2 d+3 e x)-A c e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B c \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.007, size = 100, normalized size = 0.9 \[ -{\frac{-2\,Bc{x}^{3}{e}^{3}-6\,Ac{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}-24\,Acd{e}^{2}x+6\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+2\,aA{e}^{3}-16\,Ac{d}^{2}e+4\,aBd{e}^{2}+32\,Bc{d}^{3}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)/(e*x+d)^(5/2),x)
[Out]
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Maxima [A] time = 0.69128, size = 146, normalized size = 1.3 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B c - 3 \,{\left (3 \, B c d - A c e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3} - 3 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265191, size = 147, normalized size = 1.31 \[ \frac{2 \,{\left (B c e^{3} x^{3} - 16 \, B c d^{3} + 8 \, A c d^{2} e - 2 \, B a d e^{2} - A a e^{3} - 3 \,{\left (2 \, B c d e^{2} - A c e^{3}\right )} x^{2} - 3 \,{\left (8 \, B c d^{2} e - 4 \, A c d e^{2} + B a e^{3}\right )} x\right )}}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.44001, size = 449, normalized size = 4.01 \[ \begin{cases} - \frac{2 A a e^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 A c d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 A c d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 A c e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{4 B a d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 B a e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 B c d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 B c d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 B c d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 B c e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \frac{B c x^{4}}{4}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.293875, size = 170, normalized size = 1.52 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c e^{8} - 9 \, \sqrt{x e + d} B c d e^{8} + 3 \, \sqrt{x e + d} A c e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} B c d^{2} - B c d^{3} - 6 \,{\left (x e + d\right )} A c d e + A c d^{2} e + 3 \,{\left (x e + d\right )} B a e^{2} - B a d e^{2} + A a e^{3}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]