Optimal. Leaf size=61 \[ -\frac{2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.0690738, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 \left (a e^2+c d^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 11.3967, size = 60, normalized size = 0.98 \[ \frac{4 c d}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 c}{e^{3} \sqrt{d + e x}} - \frac{2 \left (a e^{2} + c d^{2}\right )}{5 e^{3} \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.0454209, size = 44, normalized size = 0.72 \[ -\frac{2 \left (3 a e^2+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.005, size = 41, normalized size = 0.7 \[ -{\frac{30\,c{e}^{2}{x}^{2}+40\,cdex+6\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 0.700646, size = 59, normalized size = 0.97 \[ -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c - 10 \,{\left (e x + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210603, size = 82, normalized size = 1.34 \[ -\frac{2 \,{\left (15 \, c e^{2} x^{2} + 20 \, c d e x + 8 \, c d^{2} + 3 \, a e^{2}\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.35633, size = 252, normalized size = 4.13 \[ \begin{cases} - \frac{6 a e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.21074, size = 61, normalized size = 1. \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} c - 10 \,{\left (x e + d\right )} c d + 3 \, c d^{2} + 3 \, a e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^(7/2),x, algorithm="giac")
[Out]