3.2.48 \(\int \frac {(A+B x) (b x+c x^2)}{x^{9/2}} \, dx\)

Optimal. Leaf size=37 \[ -\frac {2 (A c+b B)}{3 x^{3/2}}-\frac {2 A b}{5 x^{5/2}}-\frac {2 B c}{\sqrt {x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} -\frac {2 (A c+b B)}{3 x^{3/2}}-\frac {2 A b}{5 x^{5/2}}-\frac {2 B c}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^(9/2),x]

[Out]

(-2*A*b)/(5*x^(5/2)) - (2*(b*B + A*c))/(3*x^(3/2)) - (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^{9/2}} \, dx &=\int \left (\frac {A b}{x^{7/2}}+\frac {b B+A c}{x^{5/2}}+\frac {B c}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 A b}{5 x^{5/2}}-\frac {2 (b B+A c)}{3 x^{3/2}}-\frac {2 B c}{\sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.81 \begin {gather*} -\frac {2 (A (3 b+5 c x)+5 B x (b+3 c x))}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^(9/2),x]

[Out]

(-2*(5*B*x*(b + 3*c*x) + A*(3*b + 5*c*x)))/(15*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 31, normalized size = 0.84 \begin {gather*} -\frac {2 \left (3 A b+5 A c x+5 b B x+15 B c x^2\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^(9/2),x]

[Out]

(-2*(3*A*b + 5*b*B*x + 5*A*c*x + 15*B*c*x^2))/(15*x^(5/2))

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fricas [A]  time = 0.40, size = 27, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (15 \, B c x^{2} + 3 \, A b + 5 \, {\left (B b + A c\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^(9/2),x, algorithm="fricas")

[Out]

-2/15*(15*B*c*x^2 + 3*A*b + 5*(B*b + A*c)*x)/x^(5/2)

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giac [A]  time = 0.15, size = 27, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (15 \, B c x^{2} + 5 \, B b x + 5 \, A c x + 3 \, A b\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^(9/2),x, algorithm="giac")

[Out]

-2/15*(15*B*c*x^2 + 5*B*b*x + 5*A*c*x + 3*A*b)/x^(5/2)

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maple [A]  time = 0.05, size = 28, normalized size = 0.76 \begin {gather*} -\frac {2 \left (15 B c \,x^{2}+5 A c x +5 B b x +3 A b \right )}{15 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^(9/2),x)

[Out]

-2/15/x^(5/2)*(15*B*c*x^2+5*A*c*x+5*B*b*x+3*A*b)

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maxima [A]  time = 0.51, size = 27, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (15 \, B c x^{2} + 3 \, A b + 5 \, {\left (B b + A c\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^(9/2),x, algorithm="maxima")

[Out]

-2/15*(15*B*c*x^2 + 3*A*b + 5*(B*b + A*c)*x)/x^(5/2)

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mupad [B]  time = 0.03, size = 28, normalized size = 0.76 \begin {gather*} -\frac {2\,B\,c\,x^2+\left (\frac {2\,A\,c}{3}+\frac {2\,B\,b}{3}\right )\,x+\frac {2\,A\,b}{5}}{x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^(9/2),x)

[Out]

-((2*A*b)/5 + x*((2*A*c)/3 + (2*B*b)/3) + 2*B*c*x^2)/x^(5/2)

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sympy [A]  time = 2.93, size = 46, normalized size = 1.24 \begin {gather*} - \frac {2 A b}{5 x^{\frac {5}{2}}} - \frac {2 A c}{3 x^{\frac {3}{2}}} - \frac {2 B b}{3 x^{\frac {3}{2}}} - \frac {2 B c}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**(9/2),x)

[Out]

-2*A*b/(5*x**(5/2)) - 2*A*c/(3*x**(3/2)) - 2*B*b/(3*x**(3/2)) - 2*B*c/sqrt(x)

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