Optimal. Leaf size=811 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]
[Out]
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Rubi [A] time = 6.223, antiderivative size = 811, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)/(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)/(c*x**2+a)**2,x)
[Out]
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Mathematica [C] time = 0.456171, size = 271, normalized size = 0.33 \[ \frac{\frac{2 \sqrt{a} c \sqrt{d+e x} \left (c d^2 x-a e (2 d+e x)\right )}{a+c x^2}+\frac{\sqrt{c} \left (3 \sqrt{a} e-2 i \sqrt{c} d\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{\sqrt{c} \left (\sqrt{c} d+i \sqrt{a} e\right )^2 \left (3 \sqrt{a} e+2 i \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{4 a^{3/2} c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)/(a + c*x^2)^2,x]
[Out]
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Maple [B] time = 0.217, size = 6355, normalized size = 7.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)/(c*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(c*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26698, size = 1867, normalized size = 2.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(c*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(5/2)/(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(c*x^2 + a)^2,x, algorithm="giac")
[Out]