3.2363 \(\int \frac{(d+e x)^3}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{24 c^3}+\frac{e (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c} \]

[Out]

(e*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c) + (e*(64*c^2*d^2 + 15*b^2*e^2 - 2*c*
e*(27*b*d + 8*a*e) + 10*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(24*c^3) + (
(2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x
)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(7/2))

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Rubi [A]  time = 0.408046, antiderivative size = 174, 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 \[ \frac{(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{24 c^3}+\frac{e (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(e*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c) + (e*(64*c^2*d^2 + 15*b^2*e^2 - 2*c*
e*(27*b*d + 8*a*e) + 10*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(24*c^3) + (
(2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x
)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(7/2))

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Rubi in Sympy [A]  time = 50.0329, size = 177, normalized size = 1.02 \[ \frac{e \left (d + e x\right )^{2} \sqrt{a + b x + c x^{2}}}{3 c} + \frac{e \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + \frac{15 b^{2} e^{2}}{4} - \frac{27 b c d e}{2} + 16 c^{2} d^{2} - \frac{5 c e x \left (b e - 2 c d\right )}{2}\right )}{6 c^{3}} - \frac{\left (b e - 2 c d\right ) \left (- 12 a c e^{2} + 5 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

e*(d + e*x)**2*sqrt(a + b*x + c*x**2)/(3*c) + e*sqrt(a + b*x + c*x**2)*(-4*a*c*e
**2 + 15*b**2*e**2/4 - 27*b*c*d*e/2 + 16*c**2*d**2 - 5*c*e*x*(b*e - 2*c*d)/2)/(6
*c**3) - (b*e - 2*c*d)*(-12*a*c*e**2 + 5*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*at
anh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(16*c**(7/2))

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Mathematica [A]  time = 0.22012, size = 150, normalized size = 0.86 \[ \frac{2 \sqrt{c} e \sqrt{a+x (b+c x)} \left (-2 c e (8 a e+27 b d+5 b e x)+15 b^2 e^2+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{48 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*e*Sqrt[a + x*(b + c*x)]*(15*b^2*e^2 - 2*c*e*(27*b*d + 8*a*e + 5*b*e*x
) + 4*c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*(2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e
^2 - 4*c*e*(2*b*d + 3*a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(4
8*c^(7/2))

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Maple [B]  time = 0.015, size = 366, normalized size = 2.1 \[{{d}^{3}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,b{e}^{3}x}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{2}{e}^{3}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{3}{e}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,ab{e}^{3}}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{2\,a{e}^{3}}{3\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,d{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{9\,d{e}^{2}b}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{9\,{b}^{2}d{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{3\,ad{e}^{2}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+3\,{\frac{{d}^{2}e\sqrt{c{x}^{2}+bx+a}}{c}}-{\frac{3\,{d}^{2}eb}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+b*x+a)^(1/2),x)

[Out]

d^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+1/3*e^3*x^2/c*(c*x^2+b*x
+a)^(1/2)-5/12*e^3*b/c^2*x*(c*x^2+b*x+a)^(1/2)+5/8*e^3*b^2/c^3*(c*x^2+b*x+a)^(1/
2)-5/16*e^3*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/4*e^3*b/c^
(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/3*e^3*a/c^2*(c*x^2+b*x+a)^
(1/2)+3/2*d*e^2*x/c*(c*x^2+b*x+a)^(1/2)-9/4*d*e^2*b/c^2*(c*x^2+b*x+a)^(1/2)+9/8*
d*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/2*d*e^2*a/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)-3/2*
d^2*e*b/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((e*x + d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.348064, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, c^{2} e^{3} x^{2} + 72 \, c^{2} d^{2} e - 54 \, b c d e^{2} +{\left (15 \, b^{2} - 16 \, a c\right )} e^{3} + 2 \,{\left (18 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \,{\left (3 \, b^{2} c - 4 \, a c^{2}\right )} d e^{2} -{\left (5 \, b^{3} - 12 \, a b c\right )} e^{3}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{96 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (8 \, c^{2} e^{3} x^{2} + 72 \, c^{2} d^{2} e - 54 \, b c d e^{2} +{\left (15 \, b^{2} - 16 \, a c\right )} e^{3} + 2 \,{\left (18 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 3 \,{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \,{\left (3 \, b^{2} c - 4 \, a c^{2}\right )} d e^{2} -{\left (5 \, b^{3} - 12 \, a b c\right )} e^{3}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{48 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*c^2*e^3*x^2 + 72*c^2*d^2*e - 54*b*c*d*e^2 + (15*b^2 - 16*a*c)*e^3 +
2*(18*c^2*d*e^2 - 5*b*c*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(16*c^3*d^3 -
24*b*c^2*d^2*e + 6*(3*b^2*c - 4*a*c^2)*d*e^2 - (5*b^3 - 12*a*b*c)*e^3)*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^(7/2), 1/48*(2*(8*c^2*e^3*x^2 + 72*c^2*d^2*e - 54*b*c*d*e^2 + (15*b^2 - 16
*a*c)*e^3 + 2*(18*c^2*d*e^2 - 5*b*c*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(
16*c^3*d^3 - 24*b*c^2*d^2*e + 6*(3*b^2*c - 4*a*c^2)*d*e^2 - (5*b^3 - 12*a*b*c)*e
^3)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224791, size = 230, normalized size = 1.32 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (2 \, x{\left (\frac{4 \, x e^{3}}{c} + \frac{18 \, c^{2} d e^{2} - 5 \, b c e^{3}}{c^{3}}\right )} + \frac{72 \, c^{2} d^{2} e - 54 \, b c d e^{2} + 15 \, b^{2} e^{3} - 16 \, a c e^{3}}{c^{3}}\right )} - \frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 24 \, a c^{2} d e^{2} - 5 \, b^{3} e^{3} + 12 \, a b c e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x + a)*(2*x*(4*x*e^3/c + (18*c^2*d*e^2 - 5*b*c*e^3)/c^3) + (
72*c^2*d^2*e - 54*b*c*d*e^2 + 15*b^2*e^3 - 16*a*c*e^3)/c^3) - 1/16*(16*c^3*d^3 -
24*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 24*a*c^2*d*e^2 - 5*b^3*e^3 + 12*a*b*c*e^3)*ln
(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)