Optimal. Leaf size=244 \[ \frac{e \sqrt{a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]
[Out]
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Rubi [A] time = 0.641721, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{e \sqrt{a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 83.4435, size = 221, normalized size = 0.91 \[ - \frac{5 c d e^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} + \frac{a e + c d x}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{e \sqrt{a + c x^{2}} \left (8 a^{2} e^{4} - 9 a c d^{2} e^{2} - 2 c^{2} d^{4}\right )}{3 a^{2} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{3}} + \frac{a e \left (4 a e^{2} - c d^{2}\right ) + c d x \left (7 a e^{2} + 2 c d^{2}\right )}{3 a^{2} \sqrt{a + c x^{2}} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 1.133, size = 222, normalized size = 0.91 \[ \frac{\sqrt{a+c x^2} \left (\frac{c \left (a^2 e^3 (12 d-5 e x)+9 a c d^2 e^2 x+2 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{c \left (a e^2+c d^2\right ) \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^2}-\frac{3 e^5}{d+e x}\right )}{3 \left (a e^2+c d^2\right )^3}-\frac{5 c d e^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{5 c d e^4 \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]
[Out]
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Maple [B] time = 0.018, size = 667, normalized size = 2.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*x^2+a)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.543307, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(c*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^2),x, algorithm="giac")
[Out]