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

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

[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]

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Rubi [A]  time = 0.0410241, antiderivative size = 41, 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 \[ -\frac{2 a A}{5 x^{5/2}}-\frac{2 a B}{3 x^{3/2}}-\frac{2 A c}{\sqrt{x}}+2 B c \sqrt{x} \]

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]

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Rubi in Sympy [A]  time = 5.06087, size = 42, normalized size = 1.02 \[ - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A c}{\sqrt{x}} - \frac{2 B a}{3 x^{\frac{3}{2}}} + 2 B c \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.0196687, size = 32, normalized size = 0.78 \[ -\frac{2 \left (a (3 A+5 B x)+15 c x^2 (A-B x)\right )}{15 x^{5/2}} \]

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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Maple [A]  time = 0.006, size = 30, normalized size = 0.7 \[ -{\frac{-30\,Bc{x}^{3}+30\,Ac{x}^{2}+10\,aBx+6\,aA}{15}{x}^{-{\frac{5}{2}}}} \]

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 = 0.678239, size = 41, normalized size = 1. \[ 2 \, B c \sqrt{x} - \frac{2 \,{\left (15 \, A c x^{2} + 5 \, B a x + 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + 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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Fricas [A]  time = 0.282929, size = 39, normalized size = 0.95 \[ \frac{2 \,{\left (15 \, B c x^{3} - 15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + 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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Sympy [A]  time = 4.64351, size = 42, normalized size = 1.02 \[ - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A c}{\sqrt{x}} - \frac{2 B a}{3 x^{\frac{3}{2}}} + 2 B c \sqrt{x} \]

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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GIAC/XCAS [A]  time = 0.270711, size = 41, normalized size = 1. \[ 2 \, B c \sqrt{x} - \frac{2 \,{\left (15 \, A c x^{2} + 5 \, B a x + 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + 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)