3.1615 \(\int \frac{b+2 c x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=175 \[ -\frac{2 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{2 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

[Out]

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

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Rubi [A]  time = 0.486207, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{2 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{2 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 89.3317, size = 165, normalized size = 0.94 \[ \frac{2 \sqrt{2} \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} + \frac{2 \sqrt{2} \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(2)*sqrt(c)*atan(sqrt(2)*sqrt(c)*sqrt(d + e*x)/sqrt(b*e - 2*c*d + e*sqrt(-
4*a*c + b**2)))/sqrt(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)) + 2*sqrt(2)*sqrt(c)*at
an(sqrt(2)*sqrt(c)*sqrt(d + e*x)/sqrt(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)))/sqrt
(b*e - 2*c*d - e*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.460793, size = 165, normalized size = 0.94 \[ 2 \sqrt{2} \sqrt{c} \left (-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, size = 158, normalized size = 0.9 \[ -2\,{\frac{c\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }+2\,{\frac{c\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(
1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))+2*c*2^(1/2)/((b*e-
2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*
c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*c*x + b)/((c*x^2 + b*x + a)*sqrt(e*x + d)), x)

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Fricas [A]  time = 0.292135, size = 1778, normalized size = 10.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*sqrt((2*c*d - b*e + (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/
(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))/(c*d^2
- b*d*e + a*e^2))*log(sqrt(2)*(2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sqrt((b^2
- 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*
e^2)))*sqrt((2*c*d - b*e + (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d
^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d
*e + a*e^2)) + 4*sqrt(e*x + d)*c) + 1/2*sqrt(2)*sqrt((2*c*d - b*e + (c*d^2 - b*d
*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^
4 + (b^2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^2))*log(-sqrt(2)*(2*c*d - b*e
- (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*
d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))*sqrt((2*c*d - b*e + (c*d^2 - b*d*e +
a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (
b^2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^2)) + 4*sqrt(e*x + d)*c) - 1/2*sqrt
(2)*sqrt((2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4
- 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e
+ a*e^2))*log(sqrt(2)*(2*c*d - b*e + (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*
e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))*sq
rt((2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*
c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^
2)) + 4*sqrt(e*x + d)*c) + 1/2*sqrt(2)*sqrt((2*c*d - b*e - (c*d^2 - b*d*e + a*e^
2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2
+ 2*a*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^2))*log(-sqrt(2)*(2*c*d - b*e + (c*d^2
- b*d*e + a*e^2)*sqrt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a
^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)))*sqrt((2*c*d - b*e - (c*d^2 - b*d*e + a*e^2)*sq
rt((b^2 - 4*a*c)*e^2/(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a
*c)*d^2*e^2)))/(c*d^2 - b*d*e + a*e^2)) + 4*sqrt(e*x + d)*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out