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

Optimal. Leaf size=53 \[ -\frac{2 (a B+A b)}{5 x^{5/2}}-\frac{2 a A}{7 x^{7/2}}-\frac{2 (A c+b B)}{3 x^{3/2}}-\frac{2 B c}{\sqrt{x}} \]

[Out]

(-2*a*A)/(7*x^(7/2)) - (2*(A*b + a*B))/(5*x^(5/2)) - (2*(b*B + A*c))/(3*x^(3/2)) - (2*B*c)/Sqrt[x]

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Rubi [A]  time = 0.02865, antiderivative size = 53, 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} \[ -\frac{2 (a B+A b)}{5 x^{5/2}}-\frac{2 a A}{7 x^{7/2}}-\frac{2 (A c+b B)}{3 x^{3/2}}-\frac{2 B c}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0472459, size = 45, normalized size = 0.85 \[ -\frac{2 (3 a (5 A+7 B x)+7 x (A (3 b+5 c x)+5 B x (b+3 c x)))}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*a*(5*A + 7*B*x) + 7*x*(5*B*x*(b + 3*c*x) + A*(3*b + 5*c*x))))/(105*x^(7/2))

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Maple [A]  time = 0.003, size = 42, normalized size = 0.8 \begin{align*} -{\frac{210\,Bc{x}^{3}+70\,Ac{x}^{2}+70\,Bb{x}^{2}+42\,Abx+42\,aBx+30\,aA}{105}{x}^{-{\frac{7}{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^(9/2),x)

[Out]

-2/105*(105*B*c*x^3+35*A*c*x^2+35*B*b*x^2+21*A*b*x+21*B*a*x+15*A*a)/x^(7/2)

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Maxima [A]  time = 1.14722, size = 53, normalized size = 1. \begin{align*} -\frac{2 \,{\left (105 \, B c x^{3} + 35 \,{\left (B b + A c\right )} x^{2} + 15 \, A a + 21 \,{\left (B a + A b\right )} x\right )}}{105 \, x^{\frac{7}{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^(9/2),x, algorithm="maxima")

[Out]

-2/105*(105*B*c*x^3 + 35*(B*b + A*c)*x^2 + 15*A*a + 21*(B*a + A*b)*x)/x^(7/2)

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Fricas [A]  time = 1.0071, size = 107, normalized size = 2.02 \begin{align*} -\frac{2 \,{\left (105 \, B c x^{3} + 35 \,{\left (B b + A c\right )} x^{2} + 15 \, A a + 21 \,{\left (B a + A b\right )} x\right )}}{105 \, x^{\frac{7}{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^(9/2),x, algorithm="fricas")

[Out]

-2/105*(105*B*c*x^3 + 35*(B*b + A*c)*x^2 + 15*A*a + 21*(B*a + A*b)*x)/x^(7/2)

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Sympy [A]  time = 5.02318, size = 70, normalized size = 1.32 \begin{align*} - \frac{2 A a}{7 x^{\frac{7}{2}}} - \frac{2 A b}{5 x^{\frac{5}{2}}} - \frac{2 A c}{3 x^{\frac{3}{2}}} - \frac{2 B a}{5 x^{\frac{5}{2}}} - \frac{2 B b}{3 x^{\frac{3}{2}}} - \frac{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+a)/x**(9/2),x)

[Out]

-2*A*a/(7*x**(7/2)) - 2*A*b/(5*x**(5/2)) - 2*A*c/(3*x**(3/2)) - 2*B*a/(5*x**(5/2)) - 2*B*b/(3*x**(3/2)) - 2*B*
c/sqrt(x)

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Giac [A]  time = 1.19538, size = 55, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, B b x^{2} + 35 \, A c x^{2} + 21 \, B a x + 21 \, A b x + 15 \, A a\right )}}{105 \, x^{\frac{7}{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^(9/2),x, algorithm="giac")

[Out]

-2/105*(105*B*c*x^3 + 35*B*b*x^2 + 35*A*c*x^2 + 21*B*a*x + 21*A*b*x + 15*A*a)/x^(7/2)