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]

-2/7*a*A/x^(7/2)-2/5*(A*b+B*a)/x^(5/2)-2/3*(A*c+B*b)/x^(3/2)-2*B*c/x^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, 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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fricas [A]  time = 0.52, size = 39, normalized size = 0.74 \[ -\frac {2 \, {\left (105 \, B c x^{3} + 35 \, {\left (B b + A c\right )} x^{2} + 15 \, A a + 21 \, {\left (B a + A b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \]

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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giac [A]  time = 0.15, size = 41, normalized size = 0.77 \[ -\frac {2 \, {\left (105 \, B c x^{3} + 35 \, B b x^{2} + 35 \, A c x^{2} + 21 \, B a x + 21 \, A b x + 15 \, A a\right )}}{105 \, x^{\frac {7}{2}}} \]

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)

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maple [A]  time = 0.06, size = 42, normalized size = 0.79 \[ -\frac {2 \left (105 B c \,x^{3}+35 A c \,x^{2}+35 B b \,x^{2}+21 A b x +21 B a x +15 A a \right )}{105 x^{\frac {7}{2}}} \]

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 = 0.52, size = 39, normalized size = 0.74 \[ -\frac {2 \, {\left (105 \, B c x^{3} + 35 \, {\left (B b + A c\right )} x^{2} + 15 \, A a + 21 \, {\left (B a + A b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \]

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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mupad [B]  time = 0.04, size = 41, normalized size = 0.77 \[ -\frac {2\,B\,c\,x^3+\left (\frac {2\,A\,c}{3}+\frac {2\,B\,b}{3}\right )\,x^2+\left (\frac {2\,A\,b}{5}+\frac {2\,B\,a}{5}\right )\,x+\frac {2\,A\,a}{7}}{x^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.09, size = 70, normalized size = 1.32 \[ - \frac {2 A a}{7 x^{\frac {7}{2}}} - \frac {2 A b}{5 x^{\frac {5}{2}}} - \frac {2 A c}{3 x^{\frac {3}{2}}} - \frac {2 B a}{5 x^{\frac {5}{2}}} - \frac {2 B b}{3 x^{\frac {3}{2}}} - \frac {2 B c}{\sqrt {x}} \]

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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