3.2474 \(\int \frac{(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{B e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]

[Out]

(2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*
c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (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 = 0.195417, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{B e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*
c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (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 = 17.5228, size = 119, normalized size = 0.94 \[ \frac{B e \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{2 \left (- 2 a c \left (A e + B d\right ) + b \left (A c d + B a e\right ) + x \left (B b^{2} e - b c \left (A e + B d\right ) + 2 c \left (A c d - B a e\right )\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.414554, size = 127, normalized size = 1.01 \[ \frac{\frac{2 \sqrt{c} (A c (-2 a e+b (d-e x)+2 c d x)+B (a b e-2 a c (d+e x)+b x (b e-c d)))}{\sqrt{a+x (b+c x)}}-B e \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 341, normalized size = 2.7 \[ 2\,{\frac{Ad \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{Ae}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{Bd}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{Abex}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{Bbdx}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{A{b}^{2}e}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{{b}^{2}Bd}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{Bex}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{bBe}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{2}Bex}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{3}Be}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{Be\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*d*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/c/(c*x^2+b*x+a)^(1/2)*A*e-1/c/
(c*x^2+b*x+a)^(1/2)*B*d-2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*A*e-2*b/(4*a*c-b^2
)/(c*x^2+b*x+a)^(1/2)*x*B*d-b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*A*e-b^2/c/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)*B*d-B*e*x/c/(c*x^2+b*x+a)^(1/2)+1/2*B*e/c^2*b/(c*x^2
+b*x+a)^(1/2)+B*e/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+1/2*B*e/c^2*b^3/(4*a*c
-b^2)/(c*x^2+b*x+a)^(1/2)+B*e/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(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((B*x + A)*(e*x + d)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21797, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (2 \, B a - A b\right )} c d -{\left (B a b - 2 \, A a c\right )} e +{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} +{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} e x^{2} +{\left (B b^{3} - 4 \, B a b c\right )} e x +{\left (B a b^{2} - 4 \, B a^{2} c\right )} e\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{c}}, \frac{2 \,{\left ({\left (2 \, B a - A b\right )} c d -{\left (B a b - 2 \, A a c\right )} e +{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} +{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} e x^{2} +{\left (B b^{3} - 4 \, B a b c\right )} e x +{\left (B a b^{2} - 4 \, B a^{2} c\right )} e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(4*((2*B*a - A*b)*c*d - (B*a*b - 2*A*a*c)*e + ((B*b*c - 2*A*c^2)*d - (B*b^2
- (2*B*a + A*b)*c)*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + ((B*b^2*c - 4*B*a*c^2)
*e*x^2 + (B*b^3 - 4*B*a*b*c)*e*x + (B*a*b^2 - 4*B*a^2*c)*e)*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)))/((a*b^2
*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt(c)), (2*(
(2*B*a - A*b)*c*d - (B*a*b - 2*A*a*c)*e + ((B*b*c - 2*A*c^2)*d - (B*b^2 - (2*B*a
+ A*b)*c)*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + ((B*b^2*c - 4*B*a*c^2)*e*x^2 +
(B*b^3 - 4*B*a*b*c)*e*x + (B*a*b^2 - 4*B*a^2*c)*e)*arctan(1/2*(2*c*x + b)*sqrt(
-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2
+ (b^3*c - 4*a*b*c^2)*x)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.278715, size = 198, normalized size = 1.57 \[ -\frac{B e{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} + \frac{2 \,{\left (\frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{2 \, B a c d - A b c d - B a b e + 2 \, A a c e}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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