3.434 \(\int \frac{x^2 (A+B x)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{2 a^2 (A b-a B)}{3 b^4 (a+b x)^{3/2}}+\frac{2 a (2 A b-3 a B)}{b^4 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x} (A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{3/2}}{3 b^4} \]

[Out]

(-2*a^2*(A*b - a*B))/(3*b^4*(a + b*x)^(3/2)) + (2*a*(2*A*b - 3*a*B))/(b^4*Sqrt[a
+ b*x]) + (2*(A*b - 3*a*B)*Sqrt[a + b*x])/b^4 + (2*B*(a + b*x)^(3/2))/(3*b^4)

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Rubi [A]  time = 0.120726, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a^2 (A b-a B)}{3 b^4 (a+b x)^{3/2}}+\frac{2 a (2 A b-3 a B)}{b^4 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x} (A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2*(A*b - a*B))/(3*b^4*(a + b*x)^(3/2)) + (2*a*(2*A*b - 3*a*B))/(b^4*Sqrt[a
+ b*x]) + (2*(A*b - 3*a*B)*Sqrt[a + b*x])/b^4 + (2*B*(a + b*x)^(3/2))/(3*b^4)

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Rubi in Sympy [A]  time = 16.6402, size = 88, normalized size = 0.97 \[ \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b^{4}} - \frac{2 a^{2} \left (A b - B a\right )}{3 b^{4} \left (a + b x\right )^{\frac{3}{2}}} + \frac{2 a \left (2 A b - 3 B a\right )}{b^{4} \sqrt{a + b x}} + \frac{2 \sqrt{a + b x} \left (A b - 3 B a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(a + b*x)**(3/2)/(3*b**4) - 2*a**2*(A*b - B*a)/(3*b**4*(a + b*x)**(3/2)) + 2
*a*(2*A*b - 3*B*a)/(b**4*sqrt(a + b*x)) + 2*sqrt(a + b*x)*(A*b - 3*B*a)/b**4

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Mathematica [A]  time = 0.0702667, size = 63, normalized size = 0.69 \[ \frac{2 \left (-16 a^3 B+8 a^2 b (A-3 B x)-6 a b^2 x (B x-2 A)+b^3 x^2 (3 A+B x)\right )}{3 b^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-16*a^3*B + 8*a^2*b*(A - 3*B*x) - 6*a*b^2*x*(-2*A + B*x) + b^3*x^2*(3*A + B*
x)))/(3*b^4*(a + b*x)^(3/2))

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Maple [A]  time = 0.009, size = 70, normalized size = 0.8 \[{\frac{2\,{b}^{3}B{x}^{3}+6\,A{x}^{2}{b}^{3}-12\,B{x}^{2}a{b}^{2}+24\,Axa{b}^{2}-48\,Bx{a}^{2}b+16\,A{a}^{2}b-32\,B{a}^{3}}{3\,{b}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/(b*x+a)^(3/2)*(B*b^3*x^3+3*A*b^3*x^2-6*B*a*b^2*x^2+12*A*a*b^2*x-24*B*a^2*b*x
+8*A*a^2*b-16*B*a^3)/b^4

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Maxima [A]  time = 1.3445, size = 109, normalized size = 1.2 \[ \frac{2 \,{\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}} B - 3 \,{\left (3 \, B a - A b\right )} \sqrt{b x + a}}{b} + \frac{B a^{3} - A a^{2} b - 3 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(((b*x + a)^(3/2)*B - 3*(3*B*a - A*b)*sqrt(b*x + a))/b + (B*a^3 - A*a^2*b -
3*(3*B*a^2 - 2*A*a*b)*(b*x + a))/((b*x + a)^(3/2)*b))/b^3

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Fricas [A]  time = 0.210516, size = 109, normalized size = 1.2 \[ \frac{2 \,{\left (B b^{3} x^{3} - 16 \, B a^{3} + 8 \, A a^{2} b - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{2} - 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x\right )}}{3 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(B*b^3*x^3 - 16*B*a^3 + 8*A*a^2*b - 3*(2*B*a*b^2 - A*b^3)*x^2 - 12*(2*B*a^2*
b - A*a*b^2)*x)/((b^5*x + a*b^4)*sqrt(b*x + a))

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Sympy [A]  time = 4.33084, size = 299, normalized size = 3.29 \[ \begin{cases} \frac{16 A a^{2} b}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{24 A a b^{2} x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{6 A b^{3} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{32 B a^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{48 B a^{2} b x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{12 B a b^{2} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{2 B b^{3} x^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{4}}{4}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((16*A*a**2*b/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) + 24*A*
a*b**2*x/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) + 6*A*b**3*x**2/(3*a*
b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 32*B*a**3/(3*a*b**4*sqrt(a + b*x)
+ 3*b**5*x*sqrt(a + b*x)) - 48*B*a**2*b*x/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sq
rt(a + b*x)) - 12*B*a*b**2*x**2/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)
) + 2*B*b**3*x**3/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)), Ne(b, 0)),
((A*x**3/3 + B*x**4/4)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.233547, size = 124, normalized size = 1.36 \[ -\frac{2 \,{\left (9 \,{\left (b x + a\right )} B a^{2} - B a^{3} - 6 \,{\left (b x + a\right )} A a b + A a^{2} b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{8} - 9 \, \sqrt{b x + a} B a b^{8} + 3 \, \sqrt{b x + a} A b^{9}\right )}}{3 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(9*(b*x + a)*B*a^2 - B*a^3 - 6*(b*x + a)*A*a*b + A*a^2*b)/((b*x + a)^(3/2)*
b^4) + 2/3*((b*x + a)^(3/2)*B*b^8 - 9*sqrt(b*x + a)*B*a*b^8 + 3*sqrt(b*x + a)*A*
b^9)/b^12