3.2473 \(\int \frac{(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{e \sqrt{a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c^2 \left (b^2-4 a c\right )}+\frac{2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{2 c^{5/2}} \]

[Out]

(2*(d + e*x)*(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]) + (e*(4*A*c^
2*d + 3*b^2*B*e - 2*c*(b*B*d + A*b*e + 4*a*B*e))*Sqrt[a + b*x + c*x^2])/(c^2*(b^
2 - 4*a*c)) + (e*(4*B*c*d - 3*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq
rt[a + b*x + c*x^2])])/(2*c^(5/2))

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Rubi [A]  time = 0.510994, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{e \sqrt{a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c^2 \left (b^2-4 a c\right )}+\frac{2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)*(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]) + (e*(4*A*c^
2*d + 3*b^2*B*e - 2*c*(b*B*d + A*b*e + 4*a*B*e))*Sqrt[a + b*x + c*x^2])/(c^2*(b^
2 - 4*a*c)) + (e*(4*B*c*d - 3*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq
rt[a + b*x + c*x^2])])/(2*c^(5/2))

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Rubi in Sympy [A]  time = 80.0989, size = 216, normalized size = 1.03 \[ - \frac{2 \left (d + e x\right ) \left (- 2 a c \left (A e + B d\right ) + b \left (A c d + B a e\right ) - x \left (- 2 A c^{2} d - B b^{2} e + c \left (A b e + 2 B a e + B b d\right )\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{e \sqrt{a + b x + c x^{2}} \left (- 2 A b c e + 4 A c^{2} d - 8 B a c e + 3 B b^{2} e - 2 B b c d\right )}{c^{2} \left (- 4 a c + b^{2}\right )} + \frac{e \left (2 A c e - 3 B b e + 4 B c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(d + e*x)*(-2*a*c*(A*e + B*d) + b*(A*c*d + B*a*e) - x*(-2*A*c**2*d - B*b**2*e
+ c*(A*b*e + 2*B*a*e + B*b*d)))/(c*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)) + e*
sqrt(a + b*x + c*x**2)*(-2*A*b*c*e + 4*A*c**2*d - 8*B*a*c*e + 3*B*b**2*e - 2*B*b
*c*d)/(c**2*(-4*a*c + b**2)) + e*(2*A*c*e - 3*B*b*e + 4*B*c*d)*atanh((b + 2*c*x)
/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2*c**(5/2))

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Mathematica [A]  time = 0.873431, size = 236, normalized size = 1.12 \[ \frac{\frac{2 \sqrt{c} \left (B \left (8 a^2 c e^2+a \left (-3 b^2 e^2+2 b c e (2 d+5 e x)-4 c^2 \left (d^2+2 d e x-e^2 x^2\right )\right )-b x \left (3 b^2 e^2+b c e (e x-4 d)+2 c^2 d^2\right )\right )+2 A c \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )\right )}{\sqrt{a+x (b+c x)}}+e \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) (-2 A c e+3 b B e-4 B c d)}{2 c^{5/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*(2*A*c*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*
c*e*(2*d + e*x)) + B*(8*a^2*c*e^2 - b*x*(2*c^2*d^2 + 3*b^2*e^2 + b*c*e*(-4*d + e
*x)) + a*(-3*b^2*e^2 + 2*b*c*e*(2*d + 5*e*x) - 4*c^2*(d^2 + 2*d*e*x - e^2*x^2)))
))/Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*e*(-4*B*c*d + 3*b*B*e - 2*A*c*e)*Log[b
+ 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(2*c^(5/2)*(-b^2 + 4*a*c))

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Maple [B]  time = 0.013, size = 779, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*A*d*e-2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*A*d*e+1/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*A*e^2+1/c^2*b^3/(4*a*c-b^2)
/(c*x^2+b*x+a)^(1/2)*B*d*e-3/2*B*e^2/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+2
*B*e^2*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/c/(c*x^2+b*x+a)^(1/2)*B*d^2+1
/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*e^2+3/2*B*e^2/c^2*b*x/(c*
x^2+b*x+a)^(1/2)-3/4*B*e^2/c^3*b^4/(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*d*e+1/c^2*b/(c*x^2+b*x+a)^(1/2)*B*d*e+1/2/c^2*b^3/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)*A*e^2-2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*B*d^2-b^2/c/(4*a*c-
b^2)/(c*x^2+b*x+a)^(1/2)*B*d^2+2/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*B*d*e+4
*B*e^2*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3/2*B*e^2/c^(5/2)*b*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2*B*e^2*a/c^2/(c*x^2+b*x+a)^(1/2)-2/c/(c*x^2+b*x+
a)^(1/2)*A*d*e+1/2/c^2*b/(c*x^2+b*x+a)^(1/2)*A*e^2+2/c^(3/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*B*d*e-x/c/(c*x^2+b*x+a)^(1/2)*A*e^2+2*A*d^2*(2*c*x+b)/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3/4*B*e^2/c^3*b^2/(c*x^2+b*x+a)^(1/2)+B*e^2*x^2/c
/(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)^2/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.04624, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*(2*(2*B*a - A*b)*c^2*d^2 + (B*b^2*c - 4*B*a*c^2)*e^2*x^2 - 4*(B*a*b*c -
2*A*a*c^2)*d*e + (3*B*a*b^2 - 2*(4*B*a^2 + A*a*b)*c)*e^2 + (2*(B*b*c^2 - 2*A*c^3
)*d^2 - 4*(B*b^2*c - (2*B*a + A*b)*c^2)*d*e + (3*B*b^3 + 4*A*a*c^2 - 2*(5*B*a*b
+ A*b^2)*c)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + (4*(B*a*b^2*c - 4*B*a^2*c^2)
*d*e - (3*B*a*b^3 + 8*A*a^2*c^2 - 2*(6*B*a^2*b + A*a*b^2)*c)*e^2 + (4*(B*b^2*c^2
- 4*B*a*c^3)*d*e - (3*B*b^3*c + 8*A*a*c^3 - 2*(6*B*a*b + A*b^2)*c^2)*e^2)*x^2 +
(4*(B*b^3*c - 4*B*a*b*c^2)*d*e - (3*B*b^4 + 8*A*a*b*c^2 - 2*(6*B*a*b^2 + A*b^3)
*c)*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)))/((a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^
3*c^2 - 4*a*b*c^3)*x)*sqrt(c)), 1/2*(2*(2*(2*B*a - A*b)*c^2*d^2 + (B*b^2*c - 4*B
*a*c^2)*e^2*x^2 - 4*(B*a*b*c - 2*A*a*c^2)*d*e + (3*B*a*b^2 - 2*(4*B*a^2 + A*a*b)
*c)*e^2 + (2*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(B*b^2*c - (2*B*a + A*b)*c^2)*d*e + (3*
B*b^3 + 4*A*a*c^2 - 2*(5*B*a*b + A*b^2)*c)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c
) + (4*(B*a*b^2*c - 4*B*a^2*c^2)*d*e - (3*B*a*b^3 + 8*A*a^2*c^2 - 2*(6*B*a^2*b +
A*a*b^2)*c)*e^2 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d*e - (3*B*b^3*c + 8*A*a*c^3 - 2*(
6*B*a*b + A*b^2)*c^2)*e^2)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*d*e - (3*B*b^4 + 8*A
*a*b*c^2 - 2*(6*B*a*b^2 + A*b^3)*c)*e^2)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqr
t(c*x^2 + b*x + a)*c)))/((a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3
*c^2 - 4*a*b*c^3)*x)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28201, size = 397, normalized size = 1.89 \[ \frac{{\left (\frac{{\left (B b^{2} c e^{2} - 4 \, B a c^{2} e^{2}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac{2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 8 \, B a c^{2} d e + 4 \, A b c^{2} d e + 3 \, B b^{3} e^{2} - 10 \, B a b c e^{2} - 2 \, A b^{2} c e^{2} + 4 \, A a c^{2} e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac{4 \, B a c^{2} d^{2} - 2 \, A b c^{2} d^{2} - 4 \, B a b c d e + 8 \, A a c^{2} d e + 3 \, B a b^{2} e^{2} - 8 \, B a^{2} c e^{2} - 2 \, A a b c e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt{c x^{2} + b x + a}} - \frac{{\left (4 \, B c d e - 3 \, B b e^{2} + 2 \, A c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(((B*b^2*c*e^2 - 4*B*a*c^2*e^2)*x/(b^2*c^2 - 4*a*c^3) + (2*B*b*c^2*d^2 - 4*A*c^3
*d^2 - 4*B*b^2*c*d*e + 8*B*a*c^2*d*e + 4*A*b*c^2*d*e + 3*B*b^3*e^2 - 10*B*a*b*c*
e^2 - 2*A*b^2*c*e^2 + 4*A*a*c^2*e^2)/(b^2*c^2 - 4*a*c^3))*x + (4*B*a*c^2*d^2 - 2
*A*b*c^2*d^2 - 4*B*a*b*c*d*e + 8*A*a*c^2*d*e + 3*B*a*b^2*e^2 - 8*B*a^2*c*e^2 - 2
*A*a*b*c*e^2)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a) - 1/2*(4*B*c*d*e - 3*B*
b*e^2 + 2*A*c*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c
^(5/2)