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

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

[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]

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Rubi [A]  time = 0.0408103, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a A}{7 x^{7/2}}-\frac{2 a B}{5 x^{5/2}}-\frac{2 A c}{3 x^{3/2}}-\frac{2 B c}{\sqrt{x}} \]

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]

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Rubi in Sympy [A]  time = 4.99965, size = 46, normalized size = 1.07 \[ - \frac{2 A a}{7 x^{\frac{7}{2}}} - \frac{2 A c}{3 x^{\frac{3}{2}}} - \frac{2 B a}{5 x^{\frac{5}{2}}} - \frac{2 B c}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.022307, size = 33, normalized size = 0.77 \[ \frac{-6 a (5 A+7 B x)-70 c x^2 (A+3 B x)}{105 x^{7/2}} \]

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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Maple [A]  time = 0.005, size = 30, normalized size = 0.7 \[ -{\frac{210\,Bc{x}^{3}+70\,Ac{x}^{2}+42\,aBx+30\,aA}{105}{x}^{-{\frac{7}{2}}}} \]

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 = 0.682698, size = 39, normalized size = 0.91 \[ -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + 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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Fricas [A]  time = 0.278582, size = 39, normalized size = 0.91 \[ -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + 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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Sympy [A]  time = 9.90434, size = 46, normalized size = 1.07 \[ - \frac{2 A a}{7 x^{\frac{7}{2}}} - \frac{2 A c}{3 x^{\frac{3}{2}}} - \frac{2 B a}{5 x^{\frac{5}{2}}} - \frac{2 B c}{\sqrt{x}} \]

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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GIAC/XCAS [A]  time = 0.268042, size = 39, normalized size = 0.91 \[ -\frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} + 21 \, B a x + 15 \, A a\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + 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)