3.2346 \(\int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=207 \[ -\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c} \]

[Out]

(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^4) -
(5*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^3) +
((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(24*c^2) + (e*(a + b*x + c*
x^2)^(7/2))/(7*c) - (5*(b^2 - 4*a*c)^3*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(9/2))

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Rubi [A]  time = 0.236197, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left (a+b x+c x^2\right )^{7/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^4) -
(5*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^3) +
((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(24*c^2) + (e*(a + b*x + c*
x^2)^(7/2))/(7*c) - (5*(b^2 - 4*a*c)^3*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt
[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(9/2))

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Rubi in Sympy [A]  time = 26.8788, size = 197, normalized size = 0.95 \[ \frac{e \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{7 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{24 c^{2}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{3}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{1024 c^{4}} + \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+b*x+a)**(5/2),x)

[Out]

e*(a + b*x + c*x**2)**(7/2)/(7*c) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**2)
**(5/2)/(24*c**2) + 5*(b + 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(3/2)/(384*c**3) - 5*(b + 2*c*x)*(-4*a*c + b**2)**2*(b*e - 2*c*d)*sqrt(a + b
*x + c*x**2)/(1024*c**4) + 5*(-4*a*c + b**2)**3*(b*e - 2*c*d)*atanh((b + 2*c*x)/
(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2048*c**(9/2))

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Mathematica [A]  time = 0.796627, size = 308, normalized size = 1.49 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^2 c^2 \left (-231 a^2 e+6 a c x (14 d+5 e x)+2 c^2 x^3 (189 d+148 e x)\right )+32 b c^3 \left (3 a^2 (77 d+19 e x)+2 a c x^2 (273 d+197 e x)+8 c^2 x^4 (35 d+29 e x)\right )+64 c^3 \left (48 a^3 e+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)+8 c^3 x^5 (7 d+6 e x)\right )+28 b^4 c (40 a e-c x (5 d+2 e x))+16 b^3 c^2 \left (c x^2 (7 d+3 e x)-14 a (10 d+3 e x)\right )-105 b^6 e+70 b^5 c (3 d+e x)\right )+105 \left (b^2-4 a c\right )^3 (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{43008 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^6*e + 70*b^5*c*(3*d + e*x) + 28*b^4*c*(
40*a*e - c*x*(5*d + 2*e*x)) + 16*b^3*c^2*(c*x^2*(7*d + 3*e*x) - 14*a*(10*d + 3*e
*x)) + 64*c^3*(48*a^3*e + 8*c^3*x^5*(7*d + 6*e*x) + 3*a^2*c*x*(77*d + 48*e*x) +
2*a*c^2*x^3*(91*d + 72*e*x)) + 16*b^2*c^2*(-231*a^2*e + 6*a*c*x*(14*d + 5*e*x) +
2*c^2*x^3*(189*d + 148*e*x)) + 32*b*c^3*(3*a^2*(77*d + 19*e*x) + 8*c^2*x^4*(35*
d + 29*e*x) + 2*a*c*x^2*(273*d + 197*e*x))) + 105*(b^2 - 4*a*c)^3*(-2*c*d + b*e)
*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(43008*c^(9/2))

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Maple [B]  time = 0.01, size = 807, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^(5/2),x)

[Out]

1/7*e*(c*x^2+b*x+a)^(7/2)/c-1/12*e*b/c*(c*x^2+b*x+a)^(5/2)*x-5/192*d/c^2*(c*x^2+
b*x+a)^(3/2)*b^3+5/16*d*(c*x^2+b*x+a)^(1/2)*x*a^2-5/32*d/c*(c*x^2+b*x+a)^(1/2)*x
*a*b^2+5/64*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a-5/48*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a
-5/32*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a^2+5/192*e*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x-15/5
12*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-15/64*d/c^(3/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a^2-5/96*e*b^2/c^2*(c*x^2+b*x+a)^
(3/2)*a-5/512*e*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x-5/64*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)
*a^2+5/48*d/c*(c*x^2+b*x+a)^(3/2)*b*a+5/256*d/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32
*d/c*(c*x^2+b*x+a)^(1/2)*b*a^2-5/64*d/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a+15/256*d/c^(
5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a+15/128*e*b^3/c^(5/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-1/24*e*b^2/c^2*(c*x^2+b*x+a)^(5/2)+5
/384*e*b^4/c^3*(c*x^2+b*x+a)^(3/2)-5/1024*e*b^6/c^4*(c*x^2+b*x+a)^(1/2)+5/2048*e
*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/512*d/c^3*(c*x^2+b*x+
a)^(1/2)*b^5+5/16*d/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-5/10
24*d/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^6+1/12*d/c*(c*x^2+b*x
+a)^(5/2)*b+5/24*d*(c*x^2+b*x+a)^(3/2)*x*a-5/96*d/c*(c*x^2+b*x+a)^(3/2)*x*b^2+5/
128*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a-5/32*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a^3+1/6*d*(c*x^2+b*x+a)^(5/2)*x

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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((c*x^2 + b*x + a)^(5/2)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310404, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d),x, algorithm="fricas")

[Out]

[1/86016*(4*(3072*c^6*e*x^6 + 256*(14*c^6*d + 29*b*c^5*e)*x^5 + 128*(70*b*c^5*d
+ (37*b^2*c^4 + 72*a*c^5)*e)*x^4 + 16*(14*(27*b^2*c^4 + 52*a*c^5)*d + (3*b^3*c^3
+ 788*a*b*c^4)*e)*x^3 + 8*(14*(b^3*c^3 + 156*a*b*c^4)*d - (7*b^4*c^2 - 60*a*b^2
*c^3 - 1152*a^2*c^4)*e)*x^2 + 14*(15*b^5*c - 160*a*b^3*c^2 + 528*a^2*b*c^3)*d -
(105*b^6 - 1120*a*b^4*c + 3696*a^2*b^2*c^2 - 3072*a^3*c^3)*e - 2*(14*(5*b^4*c^2
- 48*a*b^2*c^3 - 528*a^2*c^4)*d - (35*b^5*c - 336*a*b^3*c^2 + 912*a^2*b*c^3)*e)*
x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3
- 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e)*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^(9/2), 1/43008*(2*(3072*c^6*e*x^6 + 256*(14*c^6*d + 29*b*c^5*e)*x^5 + 128
*(70*b*c^5*d + (37*b^2*c^4 + 72*a*c^5)*e)*x^4 + 16*(14*(27*b^2*c^4 + 52*a*c^5)*d
+ (3*b^3*c^3 + 788*a*b*c^4)*e)*x^3 + 8*(14*(b^3*c^3 + 156*a*b*c^4)*d - (7*b^4*c
^2 - 60*a*b^2*c^3 - 1152*a^2*c^4)*e)*x^2 + 14*(15*b^5*c - 160*a*b^3*c^2 + 528*a^
2*b*c^3)*d - (105*b^6 - 1120*a*b^4*c + 3696*a^2*b^2*c^2 - 3072*a^3*c^3)*e - 2*(1
4*(5*b^4*c^2 - 48*a*b^2*c^3 - 528*a^2*c^4)*d - (35*b^5*c - 336*a*b^3*c^2 + 912*a
^2*b*c^3)*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 105*(2*(b^6*c - 12*a*b^4*c^2 +
48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c
^3)*e)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^4
)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((d + e*x)*(a + b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.253238, size = 601, normalized size = 2.9 \[ \frac{1}{21504} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \, c^{2} x e + \frac{14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac{70 \, b c^{7} d + 37 \, b^{2} c^{6} e + 72 \, a c^{7} e}{c^{6}}\right )} x + \frac{378 \, b^{2} c^{6} d + 728 \, a c^{7} d + 3 \, b^{3} c^{5} e + 788 \, a b c^{6} e}{c^{6}}\right )} x + \frac{14 \, b^{3} c^{5} d + 2184 \, a b c^{6} d - 7 \, b^{4} c^{4} e + 60 \, a b^{2} c^{5} e + 1152 \, a^{2} c^{6} e}{c^{6}}\right )} x - \frac{70 \, b^{4} c^{4} d - 672 \, a b^{2} c^{5} d - 7392 \, a^{2} c^{6} d - 35 \, b^{5} c^{3} e + 336 \, a b^{3} c^{4} e - 912 \, a^{2} b c^{5} e}{c^{6}}\right )} x + \frac{210 \, b^{5} c^{3} d - 2240 \, a b^{3} c^{4} d + 7392 \, a^{2} b c^{5} d - 105 \, b^{6} c^{2} e + 1120 \, a b^{4} c^{3} e - 3696 \, a^{2} b^{2} c^{4} e + 3072 \, a^{3} c^{5} e}{c^{6}}\right )} + \frac{5 \,{\left (2 \, b^{6} c d - 24 \, a b^{4} c^{2} d + 96 \, a^{2} b^{2} c^{3} d - 128 \, a^{3} c^{4} d - b^{7} e + 12 \, a b^{5} c e - 48 \, a^{2} b^{3} c^{2} e + 64 \, a^{3} b c^{3} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d),x, algorithm="giac")

[Out]

1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*c^2*x*e + (14*c^8*d + 29*b*c^7*
e)/c^6)*x + (70*b*c^7*d + 37*b^2*c^6*e + 72*a*c^7*e)/c^6)*x + (378*b^2*c^6*d + 7
28*a*c^7*d + 3*b^3*c^5*e + 788*a*b*c^6*e)/c^6)*x + (14*b^3*c^5*d + 2184*a*b*c^6*
d - 7*b^4*c^4*e + 60*a*b^2*c^5*e + 1152*a^2*c^6*e)/c^6)*x - (70*b^4*c^4*d - 672*
a*b^2*c^5*d - 7392*a^2*c^6*d - 35*b^5*c^3*e + 336*a*b^3*c^4*e - 912*a^2*b*c^5*e)
/c^6)*x + (210*b^5*c^3*d - 2240*a*b^3*c^4*d + 7392*a^2*b*c^5*d - 105*b^6*c^2*e +
1120*a*b^4*c^3*e - 3696*a^2*b^2*c^4*e + 3072*a^3*c^5*e)/c^6) + 5/2048*(2*b^6*c*
d - 24*a*b^4*c^2*d + 96*a^2*b^2*c^3*d - 128*a^3*c^4*d - b^7*e + 12*a*b^5*c*e - 4
8*a^2*b^3*c^2*e + 64*a^3*b*c^3*e)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
sqrt(c) - b))/c^(9/2)