Optimal. Leaf size=152 \[ 14 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-14 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{d (b d+2 c d x)^{7/2}}{a+b x+c x^2}+\frac{28}{3} c d^3 (b d+2 c d x)^{3/2} \]
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Rubi [A] time = 0.329326, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ 14 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-14 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{d (b d+2 c d x)^{7/2}}{a+b x+c x^2}+\frac{28}{3} c d^3 (b d+2 c d x)^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 78.7337, size = 151, normalized size = 0.99 \[ 14 c d^{\frac{9}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 14 c d^{\frac{9}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + \frac{28 c d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} - \frac{d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{a + b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.57849, size = 159, normalized size = 1.05 \[ (d (b+2 c x))^{9/2} \left (\frac{4 c \left (7 a+4 c x^2\right )-3 b^2+16 b c x}{3 (b+2 c x)^3 (a+x (b+c x))}+\frac{14 c \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{9/2}}-\frac{14 c \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{9/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^2,x]
[Out]
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Maple [B] time = 0.02, size = 693, normalized size = 4.6 \[{\frac{16\,c{d}^{3}}{3} \left ( 2\,cdx+bd \right ) ^{{\frac{3}{2}}}}+16\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{3/2}a}{4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2}}}-4\,{\frac{c{d}^{5}{b}^{2} \left ( 2\,cdx+bd \right ) ^{3/2}}{4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2}}}-14\,{\frac{{c}^{2}{d}^{5}\sqrt{2}a}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\ln \left ({\frac{2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}{2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}} \right ) }-28\,{\frac{{c}^{2}{d}^{5}\sqrt{2}a}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }+28\,{\frac{{c}^{2}{d}^{5}\sqrt{2}a}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }+{\frac{7\,c{d}^{5}\sqrt{2}{b}^{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+7\,{\frac{c{d}^{5}\sqrt{2}{b}^{2}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-7\,{\frac{c{d}^{5}\sqrt{2}{b}^{2}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^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((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238711, size = 945, normalized size = 6.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + 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((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.250783, size = 595, normalized size = 3.91 \[ -7 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d^{3} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - 7 \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d^{3} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) + \frac{7}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d^{3}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac{7}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c d^{3}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{16}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c d^{3} + \frac{4 \,{\left ({\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c d^{5} - 4 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{2} d^{5}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^2,x, algorithm="giac")
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