3.1619 \(\int \frac{(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=224 \[ -\frac{3 e \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{3 e \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{(d+e x)^{3/2}}{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.833367, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{3 e \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{3 e \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{(d+e x)^{3/2}}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 127.726, size = 214, normalized size = 0.96 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{a + b x + c x^{2}} - \frac{3 \sqrt{2} e \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{3 \sqrt{2} e \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.663045, size = 221, normalized size = 0.99 \[ -\frac{3 e \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{3 e \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{(d+e x)^{3/2}}{a+x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.045, size = 590, normalized size = 2.6 \[ -{\frac{{e}^{2}}{c{e}^{2}{x}^{2}+b{e}^{2}x+a{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{3}\sqrt{2}b}{2}{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-3\,{\frac{c{e}^{2}\sqrt{2}d}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }-{\frac{3\,{e}^{2}\sqrt{2}}{2}{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}+{\frac{3\,{e}^{3}\sqrt{2}b}{2}\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-3\,{\frac{c{e}^{2}\sqrt{2}d}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }+{\frac{3\,{e}^{2}\sqrt{2}}{2}\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e^2*(e*x+d)^(3/2)/(c*e^2*x^2+b*e^2*x+a*e^2)+3/2*e^3/(-e^2*(4*a*c-b^2))^(1/2)*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))*b-3*e^2*c/(-e^2*(4*a*c-b^
2))^(1/2)*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))*d-3/2*e^2*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))+3/2*e^3/(-e^2*(4*a*c-b^2))
^(1/2)*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))*b-3*e^2*c/(-e^2*(4*
a*c-b^2))^(1/2)*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))*d+3/2*e^2*
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. \[ \int \frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.302783, size = 1118, normalized size = 4.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2
- 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 + 27*
sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^
3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) - 3*sqr
t(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))
*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 - 27*sqrt(1/2)*s
qrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^
6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) - 3*sqrt(1/2)*(c*x
^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4
*a*c^2))/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 + 27*sqrt(1/2)*sqrt(e^6/(b^
2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2
- 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) + 3*sqrt(1/2)*(c*x^2 + b*x +
a)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b
^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 - 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a
*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))
*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) + 2*(e*x + d)^(3/2))/(c*x^2 + b*x + a)

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

[Out]

Timed out

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

[Out]

Timed out