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

Optimal. Leaf size=59 \[ -\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a (a B+2 A b)}{\sqrt{x}}+2 b \sqrt{x} (2 a B+A b)+\frac{2}{3} b^2 B x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0264712, antiderivative size = 59, 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, Rules used = {76} \[ -\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a (a B+2 A b)}{\sqrt{x}}+2 b \sqrt{x} (2 a B+A b)+\frac{2}{3} b^2 B x^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{x^{5/2}} \, dx &=\int \left (\frac{a^2 A}{x^{5/2}}+\frac{a (2 A b+a B)}{x^{3/2}}+\frac{b (A b+2 a B)}{\sqrt{x}}+b^2 B \sqrt{x}\right ) \, dx\\ &=-\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a (2 A b+a B)}{\sqrt{x}}+2 b (A b+2 a B) \sqrt{x}+\frac{2}{3} b^2 B x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0146947, size = 47, normalized size = 0.8 \[ \frac{2 \left (a^2 (-(A+3 B x))+6 a b x (B x-A)+b^2 x^2 (3 A+B x)\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 51, normalized size = 0.9 \begin{align*} -{\frac{-2\,B{b}^{2}{x}^{3}-6\,A{b}^{2}{x}^{2}-12\,B{x}^{2}ab+12\,aAbx+6\,{a}^{2}Bx+2\,{a}^{2}A}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06069, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + 2 \,{\left (2 \, B a b + A b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.27309, size = 112, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (B b^{2} x^{3} - A a^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.83425, size = 73, normalized size = 1.24 \begin{align*} - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 A a b}{\sqrt{x}} + 2 A b^{2} \sqrt{x} - \frac{2 B a^{2}}{\sqrt{x}} + 4 B a b \sqrt{x} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2206, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + 4 \, B a b \sqrt{x} + 2 \, A b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{2} x + 6 \, A a b x + A a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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