3.1581 \(\int \frac{(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{e (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{7/2}}+\frac{e^2 \sqrt{a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{3 c^3}+\frac{8 e^2 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c}-\frac{2 (d+e x)^4}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*(d + e*x)^4)/Sqrt[a + b*x + c*x^2] + (8*e^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2
])/(3*c) + (e^2*(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])/(3*c^3) + (e*(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])])/(2*c^(7/2))

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Rubi [A]  time = 0.507806, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{e (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{7/2}}+\frac{e^2 \sqrt{a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{3 c^3}+\frac{8 e^2 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c}-\frac{2 (d+e x)^4}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^4)/Sqrt[a + b*x + c*x^2] + (8*e^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2
])/(3*c) + (e^2*(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])/(3*c^3) + (e*(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])])/(2*c^(7/2))

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Rubi in Sympy [A]  time = 115.728, size = 207, normalized size = 1.02 \[ - \frac{2 \left (d + e x\right )^{4}}{\sqrt{a + b x + c x^{2}}} + \frac{8 e^{2} \left (d + e x\right )^{2} \sqrt{a + b x + c x^{2}}}{3 c} + \frac{4 e^{2} \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + \frac{15 b^{2} e^{2}}{4} - \frac{27 b c d e}{2} + 16 c^{2} d^{2} - \frac{5 c e x \left (b e - 2 c d\right )}{2}\right )}{3 c^{3}} - \frac{e \left (b e - 2 c d\right ) \left (- 12 a c e^{2} + 5 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(e*x+d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

-2*(d + e*x)**4/sqrt(a + b*x + c*x**2) + 8*e**2*(d + e*x)**2*sqrt(a + b*x + c*x*
*2)/(3*c) + 4*e**2*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)/(3*c**3) - e*(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)*atanh((b + 2*c*x)/(2*sqrt(c)*
sqrt(a + b*x + c*x**2)))/(2*c**(7/2))

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Mathematica [A]  time = 0.676415, size = 246, normalized size = 1.22 \[ \frac{-c e^3 \left (16 a^2 e+a b (54 d+26 e x)+b^2 x (54 d-5 e x)\right )+15 b^2 e^4 (a+b x)+2 c^2 e^2 \left (2 a \left (18 d^2+9 d e x-2 e^2 x^2\right )-b x \left (-36 d^2+9 d e x+e^2 x^2\right )\right )+c^3 \left (-6 d^4-24 d^3 e x+36 d^2 e^2 x^2+12 d e^3 x^3+2 e^4 x^4\right )}{3 c^3 \sqrt{a+x (b+c x)}}+\frac{e (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(15*b^2*e^4*(a + b*x) + c^3*(-6*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 12*d*e^3*x^3
+ 2*e^4*x^4) - c*e^3*(16*a^2*e + b^2*x*(54*d - 5*e*x) + a*b*(54*d + 26*e*x)) +
2*c^2*e^2*(2*a*(18*d^2 + 9*d*e*x - 2*e^2*x^2) - b*x*(-36*d^2 + 9*d*e*x + e^2*x^2
)))/(3*c^3*Sqrt[a + x*(b + c*x)]) + (e*(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)]])/(2*c^(7/2
))

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Maple [B]  time = 0.021, size = 1320, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^4/(c*x^2+b*x+a)^(3/2),x)

[Out]

-8/3/c*e^4*a*x^2/(c*x^2+b*x+a)^(1/2)+12*a/c*x/(c*x^2+b*x+a)^(1/2)*d*e^3-2/3*x^3/
c/(c*x^2+b*x+a)^(1/2)*b*e^4-4*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c*d^4-16/3/c^2
*e^4*a^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+12*b/c*x/(c*x^2+b*x+a)^(1/2)*d^2*e^
2-6*b^4/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d^2*e^2+9/2*b^5/c^3/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)*d*e^3-6*b/c*x^2/(c*x^2+b*x+a)^(1/2)*d*e^3-9*b^2/c^2*x/(c*x^2+b*x
+a)^(1/2)*d*e^3-5/2*b^5/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^4-18*b/c^2*a/(c*
x^2+b*x+a)^(1/2)*d*e^3+19/3*b^4/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^4-6*a/c^
2*x/(c*x^2+b*x+a)^(1/2)*b*e^4+2/3*e^4*x^4/(c*x^2+b*x+a)^(1/2)+9/2*b^3/c^3/(c*x^2
+b*x+a)^(1/2)*d*e^3-5/4*b^6/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^4+19/3*b^2/c^3
*a/(c*x^2+b*x+a)^(1/2)*e^4+5/3*b^2/c^2*x^2/(c*x^2+b*x+a)^(1/2)*e^4+9*b^2/c^(5/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^3-12*a/c^(3/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^3-12*b^3/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d^2
*e^2+48*a*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d^2*e^2-2*d^4/(c*x^2+b*x+a)^(1/2)-
2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d^4+4*d*e^3*x^3/(c*x^2+b*x+a)^(1/2)+8/c^(1
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^3*e-16/3/c^2*e^4*a^2/(c*x^2+b*
x+a)^(1/2)+12*d^2*e^2*x^2/(c*x^2+b*x+a)^(1/2)-8*d^3*e*x/(c*x^2+b*x+a)^(1/2)-5/4*
b^4/c^4/(c*x^2+b*x+a)^(1/2)*e^4-5/2*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*e^4-6*b^2/c^2/(c*x^2+b*x+a)^(1/2)*d^2*e^2-12*b/c^(3/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e^2+24*a/c/(c*x^2+b*x+a)^(1/2)*d^2*e^2+2*b*d^
4*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+6*a/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*b*e^4+5/2*b^3/c^3*x/(c*x^2+b*x+a)^(1/2)*e^4+24*a/c*b^2/(4*a*
c-b^2)/(c*x^2+b*x+a)^(1/2)*d^2*e^2-32/3/c*e^4*a^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)*x+38/3*b^3/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^4-18*b^3/c^2*a/(4*a*c-b
^2)/(c*x^2+b*x+a)^(1/2)*d*e^3+9*b^4/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*e^3-
36*b^2/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*e^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.610833, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(4*(2*c^3*e^4*x^4 - 6*c^3*d^4 + 72*a*c^2*d^2*e^2 - 54*a*b*c*d*e^3 + (15*a*
b^2 - 16*a^2*c)*e^4 + 2*(6*c^3*d*e^3 - b*c^2*e^4)*x^3 + (36*c^3*d^2*e^2 - 18*b*c
^2*d*e^3 + (5*b^2*c - 8*a*c^2)*e^4)*x^2 - (24*c^3*d^3*e - 72*b*c^2*d^2*e^2 + 18*
(3*b^2*c - 2*a*c^2)*d*e^3 - (15*b^3 - 26*a*b*c)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sq
rt(c) + 3*(16*a*c^3*d^3*e - 24*a*b*c^2*d^2*e^2 + 6*(3*a*b^2*c - 4*a^2*c^2)*d*e^3
- (5*a*b^3 - 12*a^2*b*c)*e^4 + (16*c^4*d^3*e - 24*b*c^3*d^2*e^2 + 6*(3*b^2*c^2
- 4*a*c^3)*d*e^3 - (5*b^3*c - 12*a*b*c^2)*e^4)*x^2 + (16*b*c^3*d^3*e - 24*b^2*c^
2*d^2*e^2 + 6*(3*b^3*c - 4*a*b*c^2)*d*e^3 - (5*b^4 - 12*a*b^2*c)*e^4)*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)))/((c^4*x^2 + b*c^3*x + a*c^3)*sqrt(c)), 1/6*(2*(2*c^3*e^4*x^4 - 6*c^3*d^4 +
72*a*c^2*d^2*e^2 - 54*a*b*c*d*e^3 + (15*a*b^2 - 16*a^2*c)*e^4 + 2*(6*c^3*d*e^3
- b*c^2*e^4)*x^3 + (36*c^3*d^2*e^2 - 18*b*c^2*d*e^3 + (5*b^2*c - 8*a*c^2)*e^4)*x
^2 - (24*c^3*d^3*e - 72*b*c^2*d^2*e^2 + 18*(3*b^2*c - 2*a*c^2)*d*e^3 - (15*b^3 -
26*a*b*c)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(16*a*c^3*d^3*e - 24*a*b*c
^2*d^2*e^2 + 6*(3*a*b^2*c - 4*a^2*c^2)*d*e^3 - (5*a*b^3 - 12*a^2*b*c)*e^4 + (16*
c^4*d^3*e - 24*b*c^3*d^2*e^2 + 6*(3*b^2*c^2 - 4*a*c^3)*d*e^3 - (5*b^3*c - 12*a*b
*c^2)*e^4)*x^2 + (16*b*c^3*d^3*e - 24*b^2*c^2*d^2*e^2 + 6*(3*b^3*c - 4*a*b*c^2)*
d*e^3 - (5*b^4 - 12*a*b^2*c)*e^4)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2
+ b*x + a)*c)))/((c^4*x^2 + b*c^3*x + a*c^3)*sqrt(-c))]

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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)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293242, size = 736, normalized size = 3.64 \[ \frac{{\left ({\left (2 \,{\left (\frac{{\left (b^{2} c^{3} e^{4} - 4 \, a c^{4} e^{4}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac{6 \, b^{2} c^{3} d e^{3} - 24 \, a c^{4} d e^{3} - b^{3} c^{2} e^{4} + 4 \, a b c^{3} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac{36 \, b^{2} c^{3} d^{2} e^{2} - 144 \, a c^{4} d^{2} e^{2} - 18 \, b^{3} c^{2} d e^{3} + 72 \, a b c^{3} d e^{3} + 5 \, b^{4} c e^{4} - 28 \, a b^{2} c^{2} e^{4} + 32 \, a^{2} c^{3} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{24 \, b^{2} c^{3} d^{3} e - 96 \, a c^{4} d^{3} e - 72 \, b^{3} c^{2} d^{2} e^{2} + 288 \, a b c^{3} d^{2} e^{2} + 54 \, b^{4} c d e^{3} - 252 \, a b^{2} c^{2} d e^{3} + 144 \, a^{2} c^{3} d e^{3} - 15 \, b^{5} e^{4} + 86 \, a b^{3} c e^{4} - 104 \, a^{2} b c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{6 \, b^{2} c^{3} d^{4} - 24 \, a c^{4} d^{4} - 72 \, a b^{2} c^{2} d^{2} e^{2} + 288 \, a^{2} c^{3} d^{2} e^{2} + 54 \, a b^{3} c d e^{3} - 216 \, a^{2} b c^{2} d e^{3} - 15 \, a b^{4} e^{4} + 76 \, a^{2} b^{2} c e^{4} - 64 \, a^{3} c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}}{3 \, \sqrt{c x^{2} + b x + a}} - \frac{{\left (16 \, c^{3} d^{3} e - 24 \, b c^{2} d^{2} e^{2} + 18 \, b^{2} c d e^{3} - 24 \, a c^{2} d e^{3} - 5 \, b^{3} e^{4} + 12 \, a b c e^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

1/3*(((2*((b^2*c^3*e^4 - 4*a*c^4*e^4)*x/(b^2*c^3 - 4*a*c^4) + (6*b^2*c^3*d*e^3 -
24*a*c^4*d*e^3 - b^3*c^2*e^4 + 4*a*b*c^3*e^4)/(b^2*c^3 - 4*a*c^4))*x + (36*b^2*
c^3*d^2*e^2 - 144*a*c^4*d^2*e^2 - 18*b^3*c^2*d*e^3 + 72*a*b*c^3*d*e^3 + 5*b^4*c*
e^4 - 28*a*b^2*c^2*e^4 + 32*a^2*c^3*e^4)/(b^2*c^3 - 4*a*c^4))*x - (24*b^2*c^3*d^
3*e - 96*a*c^4*d^3*e - 72*b^3*c^2*d^2*e^2 + 288*a*b*c^3*d^2*e^2 + 54*b^4*c*d*e^3
- 252*a*b^2*c^2*d*e^3 + 144*a^2*c^3*d*e^3 - 15*b^5*e^4 + 86*a*b^3*c*e^4 - 104*a
^2*b*c^2*e^4)/(b^2*c^3 - 4*a*c^4))*x - (6*b^2*c^3*d^4 - 24*a*c^4*d^4 - 72*a*b^2*
c^2*d^2*e^2 + 288*a^2*c^3*d^2*e^2 + 54*a*b^3*c*d*e^3 - 216*a^2*b*c^2*d*e^3 - 15*
a*b^4*e^4 + 76*a^2*b^2*c*e^4 - 64*a^3*c^2*e^4)/(b^2*c^3 - 4*a*c^4))/sqrt(c*x^2 +
b*x + a) - 1/2*(16*c^3*d^3*e - 24*b*c^2*d^2*e^2 + 18*b^2*c*d*e^3 - 24*a*c^2*d*e
^3 - 5*b^3*e^4 + 12*a*b*c*e^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqr
t(c) - b))/c^(7/2)