Optimal. Leaf size=905 \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac{\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]
[Out]
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Rubi [A] time = 12.6937, antiderivative size = 905, 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 \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac{\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(7/2)/(a + c*x^2)^3,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)**(7/2)/(c*x**2+a)**3,x)
[Out]
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Mathematica [C] time = 0.676836, size = 337, normalized size = 0.37 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (-5 a^3 e^3-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+6 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2}+\frac{\left (18 \sqrt{a} \sqrt{c} d e+5 i a e^2-12 i c d^2\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{c} d+i \sqrt{a} e\right )^2 \left (18 \sqrt{a} \sqrt{c} d e-5 i a e^2+12 i c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{32 a^{5/2} c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(7/2)/(a + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.247, size = 9831, normalized size = 10.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)/(c*x^2+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/(c*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.326888, size = 2364, normalized size = 2.61 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/(c*x^2 + a)^3,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)**(7/2)/(c*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 89.9026, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/(c*x^2 + a)^3,x, algorithm="giac")
[Out]