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

Optimal. Leaf size=43 \[ -\frac {2 a A}{7 x^{7/2}}-\frac {2 a B}{5 x^{5/2}}-\frac {2 A c}{3 x^{3/2}}-\frac {2 B c}{\sqrt {x}} \]

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} -\frac {2 a A}{7 x^{7/2}}-\frac {2 a B}{5 x^{5/2}}-\frac {2 A c}{3 x^{3/2}}-\frac {2 B c}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/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*c)/Sqrt[x]

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.77 \begin {gather*} \frac {-6 a (5 A+7 B x)-70 c x^2 (A+3 B x)}{105 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-70*c*x^2*(A + 3*B*x) - 6*a*(5*A + 7*B*x))/(105*x^(7/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 33, normalized size = 0.77 \begin {gather*} -\frac {2 \left (15 a A+21 a B x+35 A c x^2+105 B c x^3\right )}{105 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 29, normalized size = 0.67 \begin {gather*} -\frac {2 \, {\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 29, normalized size = 0.67 \begin {gather*} -\frac {2 \, {\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 30, normalized size = 0.70 \begin {gather*} -\frac {2 \left (105 B c \,x^{3}+35 A c \,x^{2}+21 B a x +15 a A \right )}{105 x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 29, normalized size = 0.67 \begin {gather*} -\frac {2 \, {\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.05, size = 29, normalized size = 0.67 \begin {gather*} -\frac {210\,B\,c\,x^3+70\,A\,c\,x^2+42\,B\,a\,x+30\,A\,a}{105\,x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(30*A*a + 42*B*a*x + 70*A*c*x^2 + 210*B*c*x^3)/(105*x^(7/2))

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sympy [A]  time = 3.16, size = 46, normalized size = 1.07 \begin {gather*} - \frac {2 A a}{7 x^{\frac {7}{2}}} - \frac {2 A c}{3 x^{\frac {3}{2}}} - \frac {2 B a}{5 x^{\frac {5}{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+a)/x**(9/2),x)

[Out]

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

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