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

Optimal. Leaf size=41 \[ -\frac {2 a A}{5 x^{5/2}}-\frac {2 a B}{3 x^{3/2}}-\frac {2 A c}{\sqrt {x}}+2 B c \sqrt {x} \]

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Rubi [A]  time = 0.01, antiderivative size = 41, 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}{5 x^{5/2}}-\frac {2 a B}{3 x^{3/2}}-\frac {2 A c}{\sqrt {x}}+2 B c \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 0.78 \begin {gather*} -\frac {2 \left (a (3 A+5 B x)+15 c x^2 (A-B x)\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 33, normalized size = 0.80 \begin {gather*} \frac {2 \left (-3 a A-5 a B x-15 A c x^2+15 B c x^3\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2))/x^(7/2),x]

[Out]

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

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fricas [A]  time = 0.41, size = 29, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (15 \, B c x^{3} - 15 \, A c x^{2} - 5 \, B a x - 3 \, A a\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+a)/x^(7/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 30, normalized size = 0.73 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (15 \, A c x^{2} + 5 \, B a x + 3 \, A a\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+a)/x^(7/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 30, normalized size = 0.73 \begin {gather*} -\frac {2 \left (-15 B c \,x^{3}+15 A c \,x^{2}+5 B a x +3 a A \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+a)/x^(7/2),x)

[Out]

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

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maxima [A]  time = 0.51, size = 30, normalized size = 0.73 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (15 \, A c x^{2} + 5 \, B a x + 3 \, A a\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+a)/x^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 29, normalized size = 0.71 \begin {gather*} -\frac {-30\,B\,c\,x^3+30\,A\,c\,x^2+10\,B\,a\,x+6\,A\,a}{15\,x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)*(A + B*x))/x^(7/2),x)

[Out]

-(6*A*a + 10*B*a*x + 30*A*c*x^2 - 30*B*c*x^3)/(15*x^(5/2))

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sympy [A]  time = 1.44, size = 42, normalized size = 1.02 \begin {gather*} - \frac {2 A a}{5 x^{\frac {5}{2}}} - \frac {2 A c}{\sqrt {x}} - \frac {2 B a}{3 x^{\frac {3}{2}}} + 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**(7/2),x)

[Out]

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

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