3.2372 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=177 \[ \frac{e \sqrt{a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.455978, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{e \sqrt{a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.5895, size = 173, normalized size = 0.98 \[ \frac{2 \left (d + e x\right )^{2} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{e \sqrt{a + b x + c x^{2}} \left (- 8 a c e^{2} + 3 b^{2} e^{2} - 6 b c d e + 8 c^{2} d^{2} - 2 c e x \left (b e - 2 c d\right )\right )}{c^{2} \left (- 4 a c + b^{2}\right )} - \frac{3 e^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**2*(2*a*e - b*d + x*(b*e - 2*c*d))/((-4*a*c + b**2)*sqrt(a + b*x + c
*x**2)) + e*sqrt(a + b*x + c*x**2)*(-8*a*c*e**2 + 3*b**2*e**2 - 6*b*c*d*e + 8*c*
*2*d**2 - 2*c*e*x*(b*e - 2*c*d))/(c**2*(-4*a*c + b**2)) - 3*e**2*(b*e - 2*c*d)*a
tanh((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 = 0.619376, size = 194, normalized size = 1.1 \[ \frac{\frac{2 \sqrt{c} \left (4 c \left (2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x\right )-b^2 e^2 (3 a e+c x (e x-6 d))+2 b c \left (a e^2 (3 d+5 e x)+c d^2 (d-3 e x)\right )-3 b^3 e^3 x\right )}{\sqrt{a+x (b+c x)}}+3 e^2 \left (b^2-4 a c\right ) (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 c^{5/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*(-3*b^3*e^3*x - b^2*e^2*(3*a*e + c*x*(-6*d + e*x)) + 2*b*c*(c*d^2*(d
- 3*e*x) + a*e^2*(3*d + 5*e*x)) + 4*c*(2*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 -
3*d*e*x + e^2*x^2))))/Sqrt[a + x*(b + c*x)] + 3*(b^2 - 4*a*c)*e^2*(-2*c*d + b*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 = 0.014, size = 541, normalized size = 3.1 \[ 2\,{\frac{{d}^{3} \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,b{e}^{3}x}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{e}^{3}{b}^{2}}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{b}^{3}{e}^{3}x}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{e}^{3}{b}^{4}}{4\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,b{e}^{3}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{c{x}^{2}+bx+a}}}+4\,{\frac{ab{e}^{3}x}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{a{e}^{3}{b}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-3\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,d{e}^{2}b}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{{b}^{2}d{e}^{2}x}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,{b}^{3}d{e}^{2}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{d{e}^{2}}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }-3\,{\frac{{d}^{2}e}{c\sqrt{c{x}^{2}+bx+a}}}-6\,{\frac{{d}^{2}ebx}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-3\,{\frac{{d}^{2}e{b}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+e^3*x^2/c/(c*x^2+b*x+a)^(1/2)+3/
2*e^3*b/c^2*x/(c*x^2+b*x+a)^(1/2)-3/4*e^3*b^2/c^3/(c*x^2+b*x+a)^(1/2)-3/2*e^3*b^
3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3/4*e^3*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a
)^(1/2)-3/2*e^3*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2*e^3*a/c^
2/(c*x^2+b*x+a)^(1/2)+4*e^3*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+2*e^3*a/c^2*
b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3*d*e^2*x/c/(c*x^2+b*x+a)^(1/2)+3/2*d*e^2*b/
c^2/(c*x^2+b*x+a)^(1/2)+3*d*e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+3/2*d*e^
2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+3*d*e^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))-3*d^2*e/c/(c*x^2+b*x+a)^(1/2)-6*d^2*e*b/(4*a*c-b^2)/(c*x^2
+b*x+a)^(1/2)*x-3*d^2*e*b^2/c/(4*a*c-b^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((e*x + d)^3/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.386261, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(4*(2*b*c^2*d^3 - 12*a*c^2*d^2*e + 6*a*b*c*d*e^2 - (b^2*c - 4*a*c^2)*e^3*x
^2 - (3*a*b^2 - 8*a^2*c)*e^3 + (4*c^3*d^3 - 6*b*c^2*d^2*e + 6*(b^2*c - 2*a*c^2)*
d*e^2 - (3*b^3 - 10*a*b*c)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(2*(a*b^2*c
- 4*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3 + (2*(b^2*c^2 - 4*a*c^3)*d*e^2 - (
b^3*c - 4*a*b*c^2)*e^3)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c)*e
^3)*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*b*c^2*d^3 - 12*a*c^2*d^2*e + 6*a*b*c*d*e^2
- (b^2*c - 4*a*c^2)*e^3*x^2 - (3*a*b^2 - 8*a^2*c)*e^3 + (4*c^3*d^3 - 6*b*c^2*d^2
*e + 6*(b^2*c - 2*a*c^2)*d*e^2 - (3*b^3 - 10*a*b*c)*e^3)*x)*sqrt(c*x^2 + b*x + a
)*sqrt(-c) - 3*(2*(a*b^2*c - 4*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3 + (2*(b^
2*c^2 - 4*a*c^3)*d*e^2 - (b^3*c - 4*a*b*c^2)*e^3)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d
*e^2 - (b^4 - 4*a*b^2*c)*e^3)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(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. \[ \int \frac{\left (d + e x\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229056, size = 315, normalized size = 1.78 \[ \frac{{\left (\frac{{\left (b^{2} c e^{3} - 4 \, a c^{2} e^{3}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} - \frac{4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 12 \, a c^{2} d e^{2} - 3 \, b^{3} e^{3} + 10 \, a b c e^{3}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac{2 \, b c^{2} d^{3} - 12 \, a c^{2} d^{2} e + 6 \, a b c d e^{2} - 3 \, a b^{2} e^{3} + 8 \, a^{2} c e^{3}}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt{c x^{2} + b x + a}} - \frac{3 \,{\left (2 \, c d e^{2} - b e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(((b^2*c*e^3 - 4*a*c^2*e^3)*x/(b^2*c^2 - 4*a*c^3) - (4*c^3*d^3 - 6*b*c^2*d^2*e +
6*b^2*c*d*e^2 - 12*a*c^2*d*e^2 - 3*b^3*e^3 + 10*a*b*c*e^3)/(b^2*c^2 - 4*a*c^3))
*x - (2*b*c^2*d^3 - 12*a*c^2*d^2*e + 6*a*b*c*d*e^2 - 3*a*b^2*e^3 + 8*a^2*c*e^3)/
(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a) - 3/2*(2*c*d*e^2 - b*e^3)*ln(abs(-2*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)