Optimal. Leaf size=79 \[ -\frac{2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0889694, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 9.41954, size = 70, normalized size = 0.89 \[ - \frac{\left (d + e x\right )^{2} \left (a e - c d x\right )}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} - \frac{2 \left (a e - c d x\right ) \left (a e^{2} + c d^{2}\right )}{3 a^{2} c^{2} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0808466, size = 78, normalized size = 0.99 \[ \frac{-2 a^3 e^3-3 a^2 c e \left (d^2+e^2 x^2\right )+3 a c^2 d x \left (d^2+e^2 x^2\right )+2 c^3 d^3 x^3}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a + c*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.007, size = 83, normalized size = 1.1 \[ -{\frac{-3\,a{c}^{2}d{e}^{2}{x}^{3}-2\,{c}^{3}{d}^{3}{x}^{3}+3\,{e}^{3}{x}^{2}{a}^{2}c-3\,{d}^{3}xa{c}^{2}+2\,{a}^{3}{e}^{3}+3\,{a}^{2}c{d}^{2}e}{3\,{a}^{2}{c}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(c*x^2+a)^(5/2),x)
[Out]
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Maxima [A] time = 0.716886, size = 180, normalized size = 2.28 \[ -\frac{e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{2 \, d^{3} x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d^{3} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{d e^{2} x}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{d e^{2} x}{\sqrt{c x^{2} + a} a c} - \frac{d^{2} e}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} - \frac{2 \, a e^{3}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232366, size = 144, normalized size = 1.82 \[ -\frac{{\left (3 \, a^{2} c e^{3} x^{2} - 3 \, a c^{2} d^{3} x + 3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} -{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{3}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + a)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219109, size = 119, normalized size = 1.51 \[ \frac{{\left (\frac{3 \, d^{3}}{a} - x{\left (\frac{3 \, e^{3}}{c} - \frac{{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x}{a^{2} c^{2}}\right )}\right )} x - \frac{3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3}}{a^{2} c^{2}}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + a)^(5/2),x, algorithm="giac")
[Out]