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

Optimal. Leaf size=35 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 b B \sqrt{x} \]

[Out]

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

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Rubi [A]  time = 0.0131018, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 b B \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0101252, size = 28, normalized size = 0.8 \[ -\frac{2 (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-6\,bB{x}^{2}+6\,Abx+6\,Bax+2\,Aa}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08111, size = 36, normalized size = 1.03 \begin{align*} 2 \, B b \sqrt{x} - \frac{2 \,{\left (A a + 3 \,{\left (B 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)*(B*x+A)/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.58216, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3 \, B b x^{2} - A a - 3 \,{\left (B 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)*(B*x+A)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.832327, size = 41, normalized size = 1.17 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} - \frac{2 A b}{\sqrt{x}} - \frac{2 B a}{\sqrt{x}} + 2 B b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21156, size = 36, normalized size = 1.03 \begin{align*} 2 \, B b \sqrt{x} - \frac{2 \,{\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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