Optimal. Leaf size=123 \[ -\frac{4 c \left (a e^2+3 c d^2\right )}{e^5 \sqrt{d+e x}}+\frac{8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d \sqrt{d+e x}}{e^5} \]
[Out]
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Rubi [A] time = 0.135736, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{4 c \left (a e^2+3 c d^2\right )}{e^5 \sqrt{d+e x}}+\frac{8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d \sqrt{d+e x}}{e^5} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^2/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 25.0119, size = 119, normalized size = 0.97 \[ - \frac{8 c^{2} d \sqrt{d + e x}}{e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{5}} + \frac{8 c d \left (a e^{2} + c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 c \left (a e^{2} + 3 c d^{2}\right )}{e^{5} \sqrt{d + e x}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2}}{5 e^{5} \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**2/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.100019, size = 96, normalized size = 0.78 \[ -\frac{2 \left (3 a^2 e^4+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^2/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.01, size = 106, normalized size = 0.9 \[ -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}+80\,{c}^{2}d{x}^{3}{e}^{3}+60\,ac{e}^{4}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+80\,acd{e}^{3}x+640\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}+32\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^2/(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 0.699169, size = 163, normalized size = 1.33 \[ \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 12 \, \sqrt{e x + d} c^{2} d\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 30 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{2} - 20 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231423, size = 170, normalized size = 1.38 \[ \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} - 128 \, c^{2} d^{4} - 16 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 30 \,{\left (8 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} - 40 \,{\left (8 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.69553, size = 592, normalized size = 4.81 \[ \begin{cases} - \frac{6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} - \frac{80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} + \frac{10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt{d + e x} + 30 d e^{6} x \sqrt{d + e x} + 15 e^{7} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**2/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211841, size = 176, normalized size = 1.43 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} + 30 \,{\left (x e + d\right )}^{2} a c e^{2} - 20 \,{\left (x e + d\right )} a c d e^{2} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^(7/2),x, algorithm="giac")
[Out]