3.961 \(\int \frac{x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=280 \[ \frac{\sqrt{a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{24 c^4 \left (b^2-4 a c\right )}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac{2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{9/2}} \]

[Out]

(-2*x^3*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sqrt[a
+ b*x + c*x^2]) + ((7*b^2*B - 6*A*b*c - 16*a*B*c)*x^2*Sqrt[a + b*x + c*x^2])/(3
*c^2*(b^2 - 4*a*c)) + ((105*b^4*B - 90*A*b^3*c - 460*a*b^2*B*c + 312*a*A*b*c^2 +
256*a^2*B*c^2 - 2*c*(35*b^3*B - 30*A*b^2*c - 116*a*b*B*c + 72*a*A*c^2)*x)*Sqrt[
a + b*x + c*x^2])/(24*c^4*(b^2 - 4*a*c)) - ((35*b^3*B - 30*A*b^2*c - 60*a*b*B*c
+ 24*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(9/2
))

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Rubi [A]  time = 0.700123, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\sqrt{a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{24 c^4 \left (b^2-4 a c\right )}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac{2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x^3*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sqrt[a
+ b*x + c*x^2]) + ((7*b^2*B - 6*A*b*c - 16*a*B*c)*x^2*Sqrt[a + b*x + c*x^2])/(3
*c^2*(b^2 - 4*a*c)) + ((105*b^4*B - 90*A*b^3*c - 460*a*b^2*B*c + 312*a*A*b*c^2 +
256*a^2*B*c^2 - 2*c*(35*b^3*B - 30*A*b^2*c - 116*a*b*B*c + 72*a*A*c^2)*x)*Sqrt[
a + b*x + c*x^2])/(24*c^4*(b^2 - 4*a*c)) - ((35*b^3*B - 30*A*b^2*c - 60*a*b*B*c
+ 24*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(9/2
))

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Rubi in Sympy [A]  time = 73.0593, size = 291, normalized size = 1.04 \[ \frac{2 x^{3} \left (a \left (2 A c - B b\right ) - x \left (- A b c - 2 B a c + B b^{2}\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{x^{2} \sqrt{a + b x + c x^{2}} \left (- 6 A b c - 16 B a c + 7 B b^{2}\right )}{3 c^{2} \left (- 4 a c + b^{2}\right )} + \frac{\sqrt{a + b x + c x^{2}} \left (39 A a b c^{2} - \frac{45 A b^{3} c}{4} + 32 B a^{2} c^{2} - \frac{115 B a b^{2} c}{2} + \frac{105 B b^{4}}{8} - \frac{c x \left (72 A a c^{2} - 30 A b^{2} c - 116 B a b c + 35 B b^{3}\right )}{4}\right )}{3 c^{4} \left (- 4 a c + b^{2}\right )} - \frac{\left (24 A a c^{2} - 30 A b^{2} c - 60 B a b c + 35 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**3*(a*(2*A*c - B*b) - x*(-A*b*c - 2*B*a*c + B*b**2))/(c*(-4*a*c + b**2)*sqrt
(a + b*x + c*x**2)) + x**2*sqrt(a + b*x + c*x**2)*(-6*A*b*c - 16*B*a*c + 7*B*b**
2)/(3*c**2*(-4*a*c + b**2)) + sqrt(a + b*x + c*x**2)*(39*A*a*b*c**2 - 45*A*b**3*
c/4 + 32*B*a**2*c**2 - 115*B*a*b**2*c/2 + 105*B*b**4/8 - c*x*(72*A*a*c**2 - 30*A
*b**2*c - 116*B*a*b*c + 35*B*b**3)/4)/(3*c**4*(-4*a*c + b**2)) - (24*A*a*c**2 -
30*A*b**2*c - 60*B*a*b*c + 35*B*b**3)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x
+ c*x**2)))/(16*c**(9/2))

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Mathematica [A]  time = 1.25603, size = 227, normalized size = 0.81 \[ \frac{\sqrt{a+x (b+c x)} \left (\frac{48 \left (2 a^3 B c^2+a^2 c \left (b c (3 A+5 B x)-2 A c^2 x-4 b^2 B\right )+a b^2 \left (-b c (A+5 B x)+4 A c^2 x+b^2 B\right )+b^4 x (b B-A c)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-40 a B c+2 c x (6 A c-11 b B)-42 A b c+57 b^2 B+8 B c^2 x^2\right )}{24 c^4}-\frac{\left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{16 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(57*b^2*B - 42*A*b*c - 40*a*B*c + 2*c*(-11*b*B + 6*A*c)*x
+ 8*B*c^2*x^2 + (48*(2*a^3*B*c^2 + b^4*(b*B - A*c)*x + a*b^2*(b^2*B + 4*A*c^2*x
- b*c*(A + 5*B*x)) + a^2*c*(-4*b^2*B - 2*A*c^2*x + b*c*(3*A + 5*B*x))))/((b^2 -
4*a*c)*(a + x*(b + c*x)))))/(24*c^4) - ((35*b^3*B - 30*A*b^2*c - 60*a*b*B*c + 2
4*a*A*c^2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(16*c^(9/2))

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Maple [B]  time = 0.018, size = 800, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/2*A*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+35/16*B*b^3/c^4*x/(c*x^2+b*x
+a)^(1/2)+115/12*B*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-35/32*B*b^6/c^5/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+115/24*B*b^2/c^4*a/(c*x^2+b*x+a)^(1/2)+15/4*B*b/c
^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/3*B*a/c^2*x^2/(c*x^2+b*x+
a)^(1/2)-5/4*A*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)-15/8*A*b^2/c^3*x/(c*x^2+b*x+a)^(1/2
)-7/12*B*b/c^2*x^3/(c*x^2+b*x+a)^(1/2)+35/24*B*b^2/c^3*x^2/(c*x^2+b*x+a)^(1/2)+1
5/8*A*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/2*A*a/c^(5/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/3*B*x^4/c/(c*x^2+b*x+a)^(1/2)-35/32*
B*b^4/c^5/(c*x^2+b*x+a)^(1/2)-35/16*B*b^3/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))-8/3*B*a^2/c^3/(c*x^2+b*x+a)^(1/2)+1/2*A*x^3/c/(c*x^2+b*x+a)^(1/2)+
15/16*A*b^3/c^4/(c*x^2+b*x+a)^(1/2)-35/16*B*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)*x-8/3*B*a^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-15/4*B*b/c^3*a*x/(c*x^2+
b*x+a)^(1/2)+15/8*A*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-13/4*A*b^3/c^3*a/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+115/24*B*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2
)-16/3*B*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+15/16*A*b^5/c^4/(4*a*c-b^2)
/(c*x^2+b*x+a)^(1/2)-13/4*A*b/c^3*a/(c*x^2+b*x+a)^(1/2)+3/2*A*a/c^2*x/(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((B*x + A)*x^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.555426, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(105*B*a*b^4 + 8*(B*b^2*c^3 - 4*B*a*c^4)*x^4 - 2*(7*B*b^3*c^2 + 24*A*a*
c^4 - 2*(14*B*a*b + 3*A*b^2)*c^3)*x^3 + 8*(32*B*a^3 + 39*A*a^2*b)*c^2 + (35*B*b^
4*c + 8*(16*B*a^2 + 15*A*a*b)*c^3 - 2*(86*B*a*b^2 + 15*A*b^3)*c^2)*x^2 - 10*(46*
B*a^2*b^2 + 9*A*a*b^3)*c + (105*B*b^5 - 144*A*a^2*c^3 + 4*(122*B*a^2*b + 93*A*a*
b^2)*c^2 - 10*(53*B*a*b^3 + 9*A*b^4)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(35
*B*a*b^5 - 96*A*a^3*c^3 + 48*(5*B*a^3*b + 3*A*a^2*b^2)*c^2 + (35*B*b^5*c - 96*A*
a^2*c^4 + 48*(5*B*a^2*b + 3*A*a*b^2)*c^3 - 10*(20*B*a*b^3 + 3*A*b^4)*c^2)*x^2 -
10*(20*B*a^2*b^3 + 3*A*a*b^4)*c + (35*B*b^6 - 96*A*a^2*b*c^3 + 48*(5*B*a^2*b^2 +
3*A*a*b^3)*c^2 - 10*(20*B*a*b^4 + 3*A*b^5)*c)*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^4 - 4*a^
2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 - 4*a*b*c^5)*x)*sqrt(c)), 1/48*(2*(10
5*B*a*b^4 + 8*(B*b^2*c^3 - 4*B*a*c^4)*x^4 - 2*(7*B*b^3*c^2 + 24*A*a*c^4 - 2*(14*
B*a*b + 3*A*b^2)*c^3)*x^3 + 8*(32*B*a^3 + 39*A*a^2*b)*c^2 + (35*B*b^4*c + 8*(16*
B*a^2 + 15*A*a*b)*c^3 - 2*(86*B*a*b^2 + 15*A*b^3)*c^2)*x^2 - 10*(46*B*a^2*b^2 +
9*A*a*b^3)*c + (105*B*b^5 - 144*A*a^2*c^3 + 4*(122*B*a^2*b + 93*A*a*b^2)*c^2 - 1
0*(53*B*a*b^3 + 9*A*b^4)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(35*B*a*b^5 -
96*A*a^3*c^3 + 48*(5*B*a^3*b + 3*A*a^2*b^2)*c^2 + (35*B*b^5*c - 96*A*a^2*c^4 + 4
8*(5*B*a^2*b + 3*A*a*b^2)*c^3 - 10*(20*B*a*b^3 + 3*A*b^4)*c^2)*x^2 - 10*(20*B*a^
2*b^3 + 3*A*a*b^4)*c + (35*B*b^6 - 96*A*a^2*b*c^3 + 48*(5*B*a^2*b^2 + 3*A*a*b^3)
*c^2 - 10*(20*B*a*b^4 + 3*A*b^5)*c)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x
^2 + b*x + a)*c)))/((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4
- 4*a*b*c^5)*x)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(c*x**2+b*x+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.29195, size = 495, normalized size = 1.77 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (B b^{2} c^{3} - 4 \, B a c^{4}\right )} x}{b^{2} c^{4} - 4 \, a c^{5}} - \frac{7 \, B b^{3} c^{2} - 28 \, B a b c^{3} - 6 \, A b^{2} c^{3} + 24 \, A a c^{4}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac{35 \, B b^{4} c - 172 \, B a b^{2} c^{2} - 30 \, A b^{3} c^{2} + 128 \, B a^{2} c^{3} + 120 \, A a b c^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac{105 \, B b^{5} - 530 \, B a b^{3} c - 90 \, A b^{4} c + 488 \, B a^{2} b c^{2} + 372 \, A a b^{2} c^{2} - 144 \, A a^{2} c^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac{105 \, B a b^{4} - 460 \, B a^{2} b^{2} c - 90 \, A a b^{3} c + 256 \, B a^{3} c^{2} + 312 \, A a^{2} b c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}{24 \, \sqrt{c x^{2} + b x + a}} + \frac{{\left (35 \, B b^{3} - 60 \, B a b c - 30 \, A b^{2} c + 24 \, A a c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(((2*(4*(B*b^2*c^3 - 4*B*a*c^4)*x/(b^2*c^4 - 4*a*c^5) - (7*B*b^3*c^2 - 28*B
*a*b*c^3 - 6*A*b^2*c^3 + 24*A*a*c^4)/(b^2*c^4 - 4*a*c^5))*x + (35*B*b^4*c - 172*
B*a*b^2*c^2 - 30*A*b^3*c^2 + 128*B*a^2*c^3 + 120*A*a*b*c^3)/(b^2*c^4 - 4*a*c^5))
*x + (105*B*b^5 - 530*B*a*b^3*c - 90*A*b^4*c + 488*B*a^2*b*c^2 + 372*A*a*b^2*c^2
- 144*A*a^2*c^3)/(b^2*c^4 - 4*a*c^5))*x + (105*B*a*b^4 - 460*B*a^2*b^2*c - 90*A
*a*b^3*c + 256*B*a^3*c^2 + 312*A*a^2*b*c^2)/(b^2*c^4 - 4*a*c^5))/sqrt(c*x^2 + b*
x + a) + 1/16*(35*B*b^3 - 60*B*a*b*c - 30*A*b^2*c + 24*A*a*c^2)*ln(abs(-2*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)