3.971 \(\int \frac{x^3 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{2 \left (x \left (24 a^2 B c^2+8 a A b c^2-22 a b^2 B c+3 b^4 B\right )+a \left (16 a A c^2-20 a b B c+3 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

(-2*x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(3*c*(b^2 - 4*a*c)*(a +
b*x + c*x^2)^(3/2)) - (2*(a*(3*b^3*B - 20*a*b*B*c + 16*a*A*c^2) + (3*b^4*B - 22
*a*b^2*B*c + 8*a*A*b*c^2 + 24*a^2*B*c^2)*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x
+ c*x^2]) + (B*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(5/2)

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Rubi [A]  time = 0.352753, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 \left (x \left (24 a^2 B c^2+8 a A b c^2-22 a b^2 B c+3 b^4 B\right )+a \left (16 a A c^2-20 a b B c+3 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(A + B*x))/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(3*c*(b^2 - 4*a*c)*(a +
b*x + c*x^2)^(3/2)) - (2*(a*(3*b^3*B - 20*a*b*B*c + 16*a*A*c^2) + (3*b^4*B - 22
*a*b^2*B*c + 8*a*A*b*c^2 + 24*a^2*B*c^2)*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x
+ c*x^2]) + (B*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(5/2)

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Rubi in Sympy [A]  time = 37.55, size = 192, normalized size = 1.02 \[ \frac{B \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} + \frac{2 x^{2} \left (a \left (2 A c - B b\right ) - x \left (- A b c - 2 B a c + B b^{2}\right )\right )}{3 c \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \left (\frac{a \left (16 A a c^{2} - 20 B a b c + 3 B b^{3}\right )}{2} + x \left (4 A a b c^{2} + 12 B a^{2} c^{2} - 11 B a b^{2} c + \frac{3 B b^{4}}{2}\right )\right )}{3 c^{2} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)/(c*x**2+b*x+a)**(5/2),x)

[Out]

B*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/c**(5/2) + 2*x**2*(a*(2*
A*c - B*b) - x*(-A*b*c - 2*B*a*c + B*b**2))/(3*c*(-4*a*c + b**2)*(a + b*x + c*x*
*2)**(3/2)) - 4*(a*(16*A*a*c**2 - 20*B*a*b*c + 3*B*b**3)/2 + x*(4*A*a*b*c**2 + 1
2*B*a**2*c**2 - 11*B*a*b**2*c + 3*B*b**4/2))/(3*c**2*(-4*a*c + b**2)**2*sqrt(a +
b*x + c*x**2))

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Mathematica [A]  time = 0.640398, size = 234, normalized size = 1.24 \[ \frac{-\frac{2 \left (a^2 c (2 c (A+B x)-3 b B)+a b \left (-b c (A+4 B x)+3 A c^2 x+b^2 B\right )+b^3 x (b B-A c)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}}-\frac{2 \left (8 a^2 c^3 (3 A+4 B x)+b^3 c (10 a B-A c x)-2 a b^2 c^2 (3 A+14 B x)-4 a b c^2 (8 a B-3 A c x)+b^4 c (A+4 B x)+b^5 (-B)\right )}{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)}}+3 B \sqrt{c} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(A + B*x))/(a + b*x + c*x^2)^(5/2),x]

[Out]

((-2*(-(b^5*B) + b^4*c*(A + 4*B*x) + 8*a^2*c^3*(3*A + 4*B*x) - 2*a*b^2*c^2*(3*A
+ 14*B*x) - 4*a*b*c^2*(8*a*B - 3*A*c*x) + b^3*c*(10*a*B - A*c*x)))/((b^2 - 4*a*c
)^2*Sqrt[a + x*(b + c*x)]) - (2*(b^3*(b*B - A*c)*x + a^2*c*(-3*b*B + 2*c*(A + B*
x)) + a*b*(b^2*B + 3*A*c^2*x - b*c*(A + 4*B*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x
))^(3/2)) + 3*B*Sqrt[c]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3*c^3
)

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Maple [B]  time = 0.014, size = 860, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)/(c*x^2+b*x+a)^(5/2),x)

[Out]

B/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-A*x^2/c/(c*x^2+b*x+a)^(3/2
)+1/24*A*b^2/c^3/(c*x^2+b*x+a)^(3/2)-2/3*A*a/c^2/(c*x^2+b*x+a)^(3/2)-1/3*B*x^3/c
/(c*x^2+b*x+a)^(3/2)-1/48*B*b^3/c^4/(c*x^2+b*x+a)^(3/2)-B/c^2*x/(c*x^2+b*x+a)^(1
/2)+1/2*B/c^3*b/(c*x^2+b*x+a)^(1/2)+2*B*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)+1/12*A*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+1/8*B*b^2/c^3*x/(c*x^2+b*x+
a)^(3/2)-1/48*B*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-1/6*B*b^5/c^3/(4*a*c-b^2
)^2/(c*x^2+b*x+a)^(1/2)+1/3*B*b/c^3*a/(c*x^2+b*x+a)^(3/2)+1/2*B/c^3*b^3/(4*a*c-b
^2)/(c*x^2+b*x+a)^(1/2)+1/2*B*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+4*B*b^
2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-4*A*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)-A*b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-1/3*B*b^4/c^2/(4*a*c-b^2)^2/(c
*x^2+b*x+a)^(1/2)*x-1/4*A*b/c^2*x/(c*x^2+b*x+a)^(3/2)+1/24*A*b^4/c^3/(4*a*c-b^2)
/(c*x^2+b*x+a)^(3/2)+1/3*A*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-8*A*b*a/(4*
a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/2*B*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)+2/3*A*b^3/c
/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-1/2*A*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(
3/2)-1/24*B*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+B/c^2*b^2/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)*x+1/4*B*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)

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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^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.491863, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(4*(3*B*a^2*b^3 - 20*B*a^3*b*c + 16*A*a^3*c^2 + (4*B*b^4*c + 4*(8*B*a^2 +
3*A*a*b)*c^3 - (28*B*a*b^2 + A*b^3)*c^2)*x^3 + 3*(B*b^5 - 6*B*a*b^3*c + 2*A*a*b^
2*c^2 + 8*A*a^2*c^3)*x^2 + 6*(B*a*b^4 - 7*B*a^2*b^2*c + 4*(B*a^3 + A*a^2*b)*c^2)
*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*(B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2
+ (B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 +
16*B*a^2*b*c^3)*x^3 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*x^2 + 2*(B*a*b^5 - 8*
B*a^2*b^3*c + 16*B*a^3*b*c^2)*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^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16
*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 +
16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8
*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*sqrt(c)), -1/3*(2*(3*B*a^2*b^3 - 20*B*a^3*b*c +
16*A*a^3*c^2 + (4*B*b^4*c + 4*(8*B*a^2 + 3*A*a*b)*c^3 - (28*B*a*b^2 + A*b^3)*c^2
)*x^3 + 3*(B*b^5 - 6*B*a*b^3*c + 2*A*a*b^2*c^2 + 8*A*a^2*c^3)*x^2 + 6*(B*a*b^4 -
7*B*a^2*b^2*c + 4*(B*a^3 + A*a^2*b)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*
(B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2 + (B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^
2*c^4)*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*x^3 + (B*b^6 - 6*B*a*b
^4*c + 32*B*a^3*c^3)*x^2 + 2*(B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*x)*arcta
n(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a^2*b^4*c^2 - 8*a^3*b^2
*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*
b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^
5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*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(x**3*(B*x+A)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.289606, size = 424, normalized size = 2.24 \[ -\frac{2 \,{\left ({\left ({\left (\frac{{\left (4 \, B b^{4} c - 28 \, B a b^{2} c^{2} - A b^{3} c^{2} + 32 \, B a^{2} c^{3} + 12 \, A a b c^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (B b^{5} - 6 \, B a b^{3} c + 2 \, A a b^{2} c^{2} + 8 \, A a^{2} c^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (B a b^{4} - 7 \, B a^{2} b^{2} c + 4 \, B a^{3} c^{2} + 4 \, A a^{2} b c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \, B a^{2} b^{3} - 20 \, B a^{3} b c + 16 \, A a^{3} c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} - \frac{B{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

-2/3*((((4*B*b^4*c - 28*B*a*b^2*c^2 - A*b^3*c^2 + 32*B*a^2*c^3 + 12*A*a*b*c^3)*x
/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(B*b^5 - 6*B*a*b^3*c + 2*A*a*b^2*c^2 +
8*A*a^2*c^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 6*(B*a*b^4 - 7*B*a^2*b^2
*c + 4*B*a^3*c^2 + 4*A*a^2*b*c^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (3*B
*a^2*b^3 - 20*B*a^3*b*c + 16*A*a^3*c^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c
*x^2 + b*x + a)^(3/2) - B*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)
- b))/c^(5/2)