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

Optimal. Leaf size=51 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 \sqrt{x} (A c+b B)+\frac{2}{3} B c x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0229725, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 \sqrt{x} (A c+b B)+\frac{2}{3} B c x^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0428171, size = 42, normalized size = 0.82 \[ \frac{2 x (B x (3 b+c x)-3 A (b-c x))-2 a (A+3 B x)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 41, normalized size = 0.8 \begin{align*} -{\frac{-2\,Bc{x}^{3}-6\,Ac{x}^{2}-6\,Bb{x}^{2}+6\,Abx+6\,aBx+2\,aA}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0285, size = 53, normalized size = 1.04 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \,{\left (B b + A c\right )} \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)*(c*x^2+b*x+a)/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.03, size = 90, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (B c x^{3} + 3 \,{\left (B b + A c\right )} 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)*(c*x^2+b*x+a)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.61947, size = 63, normalized size = 1.24 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} - \frac{2 A b}{\sqrt{x}} + 2 A c \sqrt{x} - \frac{2 B a}{\sqrt{x}} + 2 B b \sqrt{x} + \frac{2 B c x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19991, size = 55, normalized size = 1.08 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, B b \sqrt{x} + 2 \, A c \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)*(c*x^2+b*x+a)/x^(5/2),x, algorithm="giac")

[Out]

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