3.939 \(\int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=203 \[ \frac{5 \left (b^2-4 a c\right )^3 (b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac{B \left (a+b x+c x^2\right )^{7/2}}{7 c} \]

[Out]

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

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Rubi [A]  time = 0.214727, antiderivative size = 203, 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 \[ \frac{5 \left (b^2-4 a c\right )^3 (b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{9/2}}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac{B \left (a+b x+c x^2\right )^{7/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.0277, size = 197, normalized size = 0.97 \[ \frac{B \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{7 c} + \frac{\left (b + 2 c x\right ) \left (2 A c - B b\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{24 c^{2}} - \frac{5 \left (b + 2 c x\right ) \left (2 A c - B b\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{3}} + \frac{5 \left (b + 2 c x\right ) \left (2 A c - B b\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{1024 c^{4}} - \frac{5 \left (2 A c - B b\right ) \left (- 4 a c + b^{2}\right )^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x + c*x**2)**(7/2)/(7*c) + (b + 2*c*x)*(2*A*c - B*b)*(a + b*x + c*x**2)
**(5/2)/(24*c**2) - 5*(b + 2*c*x)*(2*A*c - B*b)*(-4*a*c + b**2)*(a + b*x + c*x**
2)**(3/2)/(384*c**3) + 5*(b + 2*c*x)*(2*A*c - B*b)*(-4*a*c + b**2)**2*sqrt(a + b
*x + c*x**2)/(1024*c**4) - 5*(2*A*c - B*b)*(-4*a*c + b**2)**3*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 = 0.615437, size = 308, normalized size = 1.52 \[ \frac{105 \left (b^2-4 a c\right )^3 (b B-2 A c) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 b^2 c^2 \left (-231 a^2 B+6 a c x (14 A+5 B x)+2 c^2 x^3 (189 A+148 B x)\right )-32 b c^3 \left (3 a^2 (77 A+19 B x)+2 a c x^2 (273 A+197 B x)+8 c^2 x^4 (35 A+29 B x)\right )-64 c^3 \left (48 a^3 B+3 a^2 c x (77 A+48 B x)+2 a c^2 x^3 (91 A+72 B x)+8 c^3 x^5 (7 A+6 B x)\right )+28 b^4 c (c x (5 A+2 B x)-40 a B)+16 b^3 c^2 \left (14 a (10 A+3 B x)-c x^2 (7 A+3 B x)\right )-70 b^5 c (3 A+B x)+105 b^6 B\right )}{43008 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(105*b^6*B - 70*b^5*c*(3*A + B*x) + 28*b^4*c*(
-40*a*B + c*x*(5*A + 2*B*x)) + 16*b^3*c^2*(-(c*x^2*(7*A + 3*B*x)) + 14*a*(10*A +
3*B*x)) - 64*c^3*(48*a^3*B + 8*c^3*x^5*(7*A + 6*B*x) + 3*a^2*c*x*(77*A + 48*B*x
) + 2*a*c^2*x^3*(91*A + 72*B*x)) - 16*b^2*c^2*(-231*a^2*B + 6*a*c*x*(14*A + 5*B*
x) + 2*c^2*x^3*(189*A + 148*B*x)) - 32*b*c^3*(3*a^2*(77*A + 19*B*x) + 8*c^2*x^4*
(35*A + 29*B*x) + 2*a*c*x^2*(273*A + 197*B*x))) + 105*(b^2 - 4*a*c)^3*(b*B - 2*A
*c)*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 = 0.011, size = 807, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/512*B*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-5/64*B*b^2/c
^2*(c*x^2+b*x+a)^(1/2)*a^2+5/128*B*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a-5/96*A/c*(c*x^2
+b*x+a)^(3/2)*x*b^2+5/48*A/c*(c*x^2+b*x+a)^(3/2)*b*a+5/256*A/c^2*(c*x^2+b*x+a)^(
1/2)*x*b^4-5/64*A/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a+1/7*B*(c*x^2+b*x+a)^(7/2)/c-5/51
2*B*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x+5/32*A/c*(c*x^2+b*x+a)^(1/2)*b*a^2+15/256*A/c^
(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-1/12*B*b/c*(c*x^2+b*x+a)
^(5/2)*x+5/192*B*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x-5/32*A/c*(c*x^2+b*x+a)^(1/2)*x*a*
b^2+5/64*B*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a-5/48*B*b/c*(c*x^2+b*x+a)^(3/2)*x*a-5/
32*B*b/c*(c*x^2+b*x+a)^(1/2)*x*a^2-15/64*A/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))*b^2*a^2-5/96*B*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a-5/32*B*b/c^(3/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+5/512*A/c^3*(c*x^2+b*x+a)^(1/2)*b^5
+5/16*A/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-5/1024*A/c^(7/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^6+5/384*B*b^4/c^3*(c*x^2+b*x+a)^(
3/2)-5/1024*B*b^6/c^4*(c*x^2+b*x+a)^(1/2)+15/128*B*b^3/c^(5/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+1/6*A*x*(c*x^2+b*x+a)^(5/2)-1/24*B*b^2/c^2*(c*x^2
+b*x+a)^(5/2)+1/12*A/c*(c*x^2+b*x+a)^(5/2)*b+5/2048*B*b^7/c^(9/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/24*A*(c*x^2+b*x+a)^(3/2)*x*a-5/192*A/c^2*(c*x^2+
b*x+a)^(3/2)*b^3+5/16*A*(c*x^2+b*x+a)^(1/2)*x*a^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((c*x^2 + b*x + a)^(5/2)*(B*x + A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.373705, 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)*(B*x + A),x, algorithm="fricas")

[Out]

[1/86016*(4*(3072*B*c^6*x^6 - 105*B*b^6 + 256*(29*B*b*c^5 + 14*A*c^6)*x^5 + 128*
(37*B*b^2*c^4 + 2*(36*B*a + 35*A*b)*c^5)*x^4 + 96*(32*B*a^3 + 77*A*a^2*b)*c^3 +
16*(3*B*b^3*c^3 + 728*A*a*c^5 + 2*(394*B*a*b + 189*A*b^2)*c^4)*x^3 - 112*(33*B*a
^2*b^2 + 20*A*a*b^3)*c^2 - 8*(7*B*b^4*c^2 - 24*(48*B*a^2 + 91*A*a*b)*c^4 - 2*(30
*B*a*b^2 + 7*A*b^3)*c^3)*x^2 + 70*(16*B*a*b^4 + 3*A*b^5)*c + 2*(35*B*b^5*c + 739
2*A*a^2*c^4 + 48*(19*B*a^2*b + 14*A*a*b^2)*c^3 - 14*(24*B*a*b^3 + 5*A*b^4)*c^2)*
x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 105*(B*b^7 + 128*A*a^3*c^4 - 32*(2*B*a^3*b +
3*A*a^2*b^2)*c^3 + 24*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 2*(6*B*a*b^5 + A*b^6)*c)*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)*s
qrt(c)))/c^(9/2), 1/43008*(2*(3072*B*c^6*x^6 - 105*B*b^6 + 256*(29*B*b*c^5 + 14*
A*c^6)*x^5 + 128*(37*B*b^2*c^4 + 2*(36*B*a + 35*A*b)*c^5)*x^4 + 96*(32*B*a^3 + 7
7*A*a^2*b)*c^3 + 16*(3*B*b^3*c^3 + 728*A*a*c^5 + 2*(394*B*a*b + 189*A*b^2)*c^4)*
x^3 - 112*(33*B*a^2*b^2 + 20*A*a*b^3)*c^2 - 8*(7*B*b^4*c^2 - 24*(48*B*a^2 + 91*A
*a*b)*c^4 - 2*(30*B*a*b^2 + 7*A*b^3)*c^3)*x^2 + 70*(16*B*a*b^4 + 3*A*b^5)*c + 2*
(35*B*b^5*c + 7392*A*a^2*c^4 + 48*(19*B*a^2*b + 14*A*a*b^2)*c^3 - 14*(24*B*a*b^3
+ 5*A*b^4)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 105*(B*b^7 + 128*A*a^3*c^4
- 32*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 24*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 2*(6*B*a*b
^5 + A*b^6)*c)*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. \[ \int \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.28598, size = 574, normalized size = 2.83 \[ \frac{1}{21504} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \, B c^{2} x + \frac{29 \, B b c^{7} + 14 \, A c^{8}}{c^{6}}\right )} x + \frac{37 \, B b^{2} c^{6} + 72 \, B a c^{7} + 70 \, A b c^{7}}{c^{6}}\right )} x + \frac{3 \, B b^{3} c^{5} + 788 \, B a b c^{6} + 378 \, A b^{2} c^{6} + 728 \, A a c^{7}}{c^{6}}\right )} x - \frac{7 \, B b^{4} c^{4} - 60 \, B a b^{2} c^{5} - 14 \, A b^{3} c^{5} - 1152 \, B a^{2} c^{6} - 2184 \, A a b c^{6}}{c^{6}}\right )} x + \frac{35 \, B b^{5} c^{3} - 336 \, B a b^{3} c^{4} - 70 \, A b^{4} c^{4} + 912 \, B a^{2} b c^{5} + 672 \, A a b^{2} c^{5} + 7392 \, A a^{2} c^{6}}{c^{6}}\right )} x - \frac{105 \, B b^{6} c^{2} - 1120 \, B a b^{4} c^{3} - 210 \, A b^{5} c^{3} + 3696 \, B a^{2} b^{2} c^{4} + 2240 \, A a b^{3} c^{4} - 3072 \, B a^{3} c^{5} - 7392 \, A a^{2} b c^{5}}{c^{6}}\right )} - \frac{5 \,{\left (B b^{7} - 12 \, B a b^{5} c - 2 \, A b^{6} c + 48 \, B a^{2} b^{3} c^{2} + 24 \, A a b^{4} c^{2} - 64 \, B a^{3} b c^{3} - 96 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*B*c^2*x + (29*B*b*c^7 + 14*A*c^
8)/c^6)*x + (37*B*b^2*c^6 + 72*B*a*c^7 + 70*A*b*c^7)/c^6)*x + (3*B*b^3*c^5 + 788
*B*a*b*c^6 + 378*A*b^2*c^6 + 728*A*a*c^7)/c^6)*x - (7*B*b^4*c^4 - 60*B*a*b^2*c^5
- 14*A*b^3*c^5 - 1152*B*a^2*c^6 - 2184*A*a*b*c^6)/c^6)*x + (35*B*b^5*c^3 - 336*
B*a*b^3*c^4 - 70*A*b^4*c^4 + 912*B*a^2*b*c^5 + 672*A*a*b^2*c^5 + 7392*A*a^2*c^6)
/c^6)*x - (105*B*b^6*c^2 - 1120*B*a*b^4*c^3 - 210*A*b^5*c^3 + 3696*B*a^2*b^2*c^4
+ 2240*A*a*b^3*c^4 - 3072*B*a^3*c^5 - 7392*A*a^2*b*c^5)/c^6) - 5/2048*(B*b^7 -
12*B*a*b^5*c - 2*A*b^6*c + 48*B*a^2*b^3*c^2 + 24*A*a*b^4*c^2 - 64*B*a^3*b*c^3 -
96*A*a^2*b^2*c^3 + 128*A*a^3*c^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
sqrt(c) - b))/c^(9/2)