Optimal. Leaf size=159 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}+\frac{4}{d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]
[Out]
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Rubi [A] time = 0.368067, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}+\frac{4}{d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 84.8592, size = 156, normalized size = 0.98 \[ \frac{4}{5 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{4}{d^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.307427, size = 142, normalized size = 0.89 \[ \frac{2 \left (4 \sqrt [4]{b^2-4 a c} \left (2 c \left (5 c x^2-a\right )+3 b^2+10 b c x\right )+5 (b+2 c x)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 (b+2 c x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{5 d \left (b^2-4 a c\right )^{9/4} (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)),x]
[Out]
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Maple [B] time = 0.015, size = 369, normalized size = 2.3 \[ -{\frac{4}{5\,d \left ( 4\,ac-{b}^{2} \right ) } \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}}+4\,{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{2\,cdx+bd}}}+{\frac{\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{\sqrt{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{\sqrt{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a),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/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252522, size = 2037, normalized size = 12.81 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)),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(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.234334, size = 817, normalized size = 5.14 \[ -\frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{6} d^{5} - 12 \, \sqrt{2} a b^{4} c d^{5} + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt{2} a^{3} c^{3} d^{5}} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{6} d^{5} - 12 \, \sqrt{2} a b^{4} c d^{5} + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt{2} a^{3} c^{3} d^{5}} + \frac{4 \,{\left (b^{2} d^{2} - 4 \, a c d^{2} + 5 \,{\left (2 \, c d x + b d\right )}^{2}\right )}}{5 \,{\left (b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}\right )}{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)),x, algorithm="giac")
[Out]