Optimal. Leaf size=155 \[ \frac{e^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.307591, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{e^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 47.4238, size = 146, normalized size = 0.94 \[ - \frac{e^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (- 2 a c e + b^{2} e - b c d + c x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.771212, size = 207, normalized size = 1.34 \[ \frac{e^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+2 \sqrt{e (a e-b d)+c d^2} \left (2 c (a e+c d x)+b^2 (-e)+b c (d-e x)\right )+e^2 \left (-\left (b^2-4 a c\right )\right ) \sqrt{a+x (b+c x)} \log (d+e x)}{\left (4 a c-b^2\right ) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.014, size = 603, normalized size = 3.9 \[{\frac{e}{a{e}^{2}-bde+c{d}^{2}}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}-2\,{\frac{xbce}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}+4\,{\frac{x{c}^{2}d}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{b}^{2}e}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}+2\,{\frac{bcd}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{e}{a{e}^{2}-bde+c{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*x^2+b*x+a)^(3/2),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 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.578942, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)
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GIAC/XCAS [A] time = 0.222204, size = 603, normalized size = 3.89 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2} + 2 \, a c^{2} d e^{2} - a b c e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + 2 \, a c^{2} d^{2} e + b^{3} d e^{2} - a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} + \frac{2 \, \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right ) e^{2}}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="giac")
[Out]