3.1583 \(\int \frac{(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}}-\frac{2 (d+e x)^2}{\sqrt{a+b x+c x^2}}+\frac{4 e^2 \sqrt{a+b x+c x^2}}{c} \]

[Out]

(-2*(d + e*x)^2)/Sqrt[a + b*x + c*x^2] + (4*e^2*Sqrt[a + b*x + c*x^2])/c + (2*e*
(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(3/2)

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Rubi [A]  time = 0.170322, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}}-\frac{2 (d+e x)^2}{\sqrt{a+b x+c x^2}}+\frac{4 e^2 \sqrt{a+b x+c x^2}}{c} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^2)/Sqrt[a + b*x + c*x^2] + (4*e^2*Sqrt[a + b*x + c*x^2])/c + (2*e*
(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(3/2)

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Rubi in Sympy [A]  time = 40.4089, size = 87, normalized size = 0.94 \[ - \frac{2 \left (d + e x\right )^{2}}{\sqrt{a + b x + c x^{2}}} + \frac{4 e^{2} \sqrt{a + b x + c x^{2}}}{c} - \frac{2 e \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(e*x+d)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

-2*(d + e*x)**2/sqrt(a + b*x + c*x**2) + 4*e**2*sqrt(a + b*x + c*x**2)/c - 2*e*(
b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/c**(3/2)

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Mathematica [A]  time = 0.264592, size = 93, normalized size = 1. \[ \frac{2 e (2 c d-b e) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2}}+\frac{4 e^2 (a+b x)-2 c \left (d^2+2 d e x-e^2 x^2\right )}{c \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(4*e^2*(a + b*x) - 2*c*(d^2 + 2*d*e*x - e^2*x^2))/(c*Sqrt[a + x*(b + c*x)]) + (2
*e*(2*c*d - b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(3/2)

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Maple [B]  time = 0.014, size = 427, normalized size = 4.6 \[ 2\,{\frac{{d}^{2}b \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{b{e}^{2}x}{c\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{dex}{\sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{2}{e}^{2}}{{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{{b}^{3}x{e}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{4}{e}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{b{e}^{2}}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }+4\,{\frac{de}{\sqrt{c}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }-2\,{\frac{{d}^{2}}{\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{bxc{d}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{{b}^{2}{d}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{{e}^{2}{x}^{2}}{\sqrt{c{x}^{2}+bx+a}}}+4\,{\frac{a{e}^{2}}{c\sqrt{c{x}^{2}+bx+a}}}+8\,{\frac{ab{e}^{2}x}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+4\,{\frac{a{e}^{2}{b}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x)

[Out]

2*d^2*b*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+2*x/c/(c*x^2+b*x+a)^(1/2)*b*e^
2-4*d*e*x/(c*x^2+b*x+a)^(1/2)-b^2/c^2/(c*x^2+b*x+a)^(1/2)*e^2-2*b^3/c/(4*a*c-b^2
)/(c*x^2+b*x+a)^(1/2)*x*e^2-b^4/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^2-2/c^(3/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b*e^2+4/c^(1/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*d*e-2*d^2/(c*x^2+b*x+a)^(1/2)-4*b/(4*a*c-b^2)/(c*x^2+b
*x+a)^(1/2)*x*c*d^2-2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d^2+2*x^2*e^2/(c*x^2+b
*x+a)^(1/2)+4*e^2/c*a/(c*x^2+b*x+a)^(1/2)+8*e^2*a*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)*x+4*e^2/c*a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)

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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*x + b)*(e*x + d)^2/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.4769, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (c e^{2} x^{2} - c d^{2} + 2 \, a e^{2} - 2 \,{\left (c d e - b e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left (2 \, a c d e - a b e^{2} +{\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} +{\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{{\left (c^{2} x^{2} + b c x + a c\right )} \sqrt{c}}, \frac{2 \,{\left ({\left (c e^{2} x^{2} - c d^{2} + 2 \, a e^{2} - 2 \,{\left (c d e - b e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} +{\left (2 \, a c d e - a b e^{2} +{\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} +{\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )\right )}}{{\left (c^{2} x^{2} + b c x + a c\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^2/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

[(2*(c*e^2*x^2 - c*d^2 + 2*a*e^2 - 2*(c*d*e - b*e^2)*x)*sqrt(c*x^2 + b*x + a)*sq
rt(c) - (2*a*c*d*e - a*b*e^2 + (2*c^2*d*e - b*c*e^2)*x^2 + (2*b*c*d*e - b^2*e^2)
*x)*log(4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4
*a*c)*sqrt(c)))/((c^2*x^2 + b*c*x + a*c)*sqrt(c)), 2*((c*e^2*x^2 - c*d^2 + 2*a*e
^2 - 2*(c*d*e - b*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + (2*a*c*d*e - a*b*e^2
+ (2*c^2*d*e - b*c*e^2)*x^2 + (2*b*c*d*e - b^2*e^2)*x)*arctan(1/2*(2*c*x + b)*sq
rt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((c^2*x^2 + b*c*x + a*c)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b + 2 c x\right ) \left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(e*x+d)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**2/(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.290099, size = 266, normalized size = 2.86 \[ \frac{2 \,{\left ({\left (\frac{{\left (b^{2} c e^{2} - 4 \, a c^{2} e^{2}\right )} x}{b^{2} c - 4 \, a c^{2}} - \frac{2 \,{\left (b^{2} c d e - 4 \, a c^{2} d e - b^{3} e^{2} + 4 \, a b c e^{2}\right )}}{b^{2} c - 4 \, a c^{2}}\right )} x - \frac{b^{2} c d^{2} - 4 \, a c^{2} d^{2} - 2 \, a b^{2} e^{2} + 8 \, a^{2} c e^{2}}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{2 \,{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^2/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

2*(((b^2*c*e^2 - 4*a*c^2*e^2)*x/(b^2*c - 4*a*c^2) - 2*(b^2*c*d*e - 4*a*c^2*d*e -
b^3*e^2 + 4*a*b*c*e^2)/(b^2*c - 4*a*c^2))*x - (b^2*c*d^2 - 4*a*c^2*d^2 - 2*a*b^
2*e^2 + 8*a^2*c*e^2)/(b^2*c - 4*a*c^2))/sqrt(c*x^2 + b*x + a) - 2*(2*c*d*e - b*e
^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(3/2)