3.147 \(\int \frac{(A+B x) (b x+c x^2)}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0162382, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ -\frac{2 (A c+b B)}{\sqrt{x}}-\frac{2 A b}{3 x^{3/2}}+2 B c \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^(7/2),x]

[Out]

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

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 (b x+c x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac{A b}{x^{5/2}}+\frac{b B+A c}{x^{3/2}}+\frac{B c}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 A b}{3 x^{3/2}}-\frac{2 (b B+A c)}{\sqrt{x}}+2 B c \sqrt{x}\\ \end{align*}

Mathematica [A]  time = 0.0099461, size = 28, normalized size = 0.8 \[ -\frac{2 (A (b+3 c x)+3 B x (b-c x))}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-6\,Bc{x}^{2}+6\,Acx+6\,bBx+2\,Ab}{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)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.09719, size = 36, normalized size = 1.03 \begin{align*} 2 \, B c \sqrt{x} - \frac{2 \,{\left (A b + 3 \,{\left (B b + A c\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)/x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76654, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3 \, B c x^{2} - A b - 3 \,{\left (B b + A c\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)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.69926, size = 41, normalized size = 1.17 \begin{align*} - \frac{2 A b}{3 x^{\frac{3}{2}}} - \frac{2 A c}{\sqrt{x}} - \frac{2 B b}{\sqrt{x}} + 2 B c \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**(7/2),x)

[Out]

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

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Giac [A]  time = 1.11849, size = 36, normalized size = 1.03 \begin{align*} 2 \, B c \sqrt{x} - \frac{2 \,{\left (3 \, B b x + 3 \, A c x + A b\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)/x^(7/2),x, algorithm="giac")

[Out]

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