3.2466 \(\int \frac{(A+B x) (d+e x)}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=116 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c (2 A c d-a B e)-4 b c (A e+B d)+3 b^2 B e\right )}{8 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.241965, antiderivative size = 115, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 a B c e-4 b c (A e+B d)+8 A c^2 d+3 b^2 B e\right )}{8 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.2178, size = 114, normalized size = 0.98 \[ \frac{\sqrt{a + b x + c x^{2}} \left (- \frac{3 B b e}{2} + B c e x + 2 c \left (A e + B d\right )\right )}{2 c^{2}} - \frac{\left (4 B a c e - 3 B b^{2} e + 4 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.152771, size = 112, normalized size = 0.97 \[ \frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (4 c (2 A c d-a B e)-4 b c (A e+B d)+3 b^2 B e\right )}{8 c^{5/2}}+\frac{\sqrt{a+x (b+c x)} (4 A c e+B (-3 b e+4 c d+2 c e x))}{4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 243, normalized size = 2.1 \[{Ad\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{Ae}{c}\sqrt{c{x}^{2}+bx+a}}+{\frac{Bd}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{Abe}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{bBd}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{Bex}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,bBe}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}Be}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{aBe}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(1/2),x)

[Out]

A*d*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(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-1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*A*e-1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d+1/2*
B*e*x/c*(c*x^2+b*x+a)^(1/2)-3/4*B*e/c^2*b*(c*x^2+b*x+a)^(1/2)+3/8*B*e/c^(5/2)*b^
2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/2*B*e*a/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)/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.17373, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, B c e x + 4 \, B c d -{\left (3 \, B b - 4 \, A c\right )} e\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} +{\left (4 \,{\left (B b c - 2 \, A c^{2}\right )} d -{\left (3 \, B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} e\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{16 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (2 \, B c e x + 4 \, B c d -{\left (3 \, B b - 4 \, A c\right )} e\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} -{\left (4 \,{\left (B b c - 2 \, A c^{2}\right )} d -{\left (3 \, B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{8 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*B*c*e*x + 4*B*c*d - (3*B*b - 4*A*c)*e)*sqrt(c*x^2 + b*x + a)*sqrt(c)
+ (4*(B*b*c - 2*A*c^2)*d - (3*B*b^2 - 4*(B*a + A*b)*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)))/c^(5/2),
1/8*(2*(2*B*c*e*x + 4*B*c*d - (3*B*b - 4*A*c)*e)*sqrt(c*x^2 + b*x + a)*sqrt(-c)
- (4*(B*b*c - 2*A*c^2)*d - (3*B*b^2 - 4*(B*a + A*b)*c)*e)*arctan(1/2*(2*c*x + b)
*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.277439, size = 161, normalized size = 1.39 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, B x e}{c} + \frac{4 \, B c d - 3 \, B b e + 4 \, A c e}{c^{2}}\right )} + \frac{{\left (4 \, B b c d - 8 \, A c^{2} d - 3 \, B b^{2} e + 4 \, B a c e + 4 \, A b c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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