3.10.12 \(\int \frac {(A+B x) (a+b x+c x^2)}{x^{5/2}} \, dx\)

Optimal. Leaf size=51 \[ -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 \sqrt {x} (A c+b B)+\frac {2}{3} B c x^{3/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 51, 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} \begin {gather*} -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 \sqrt {x} (A c+b B)+\frac {2}{3} B c x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2))/x^(5/2),x]

[Out]

(-2*a*A)/(3*x^(3/2)) - (2*(A*b + a*B))/Sqrt[x] + 2*(b*B + A*c)*Sqrt[x] + (2*B*c*x^(3/2))/3

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

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Mathematica [A]  time = 0.05, size = 42, normalized size = 0.82 \begin {gather*} \frac {2 x (B x (3 b+c x)-3 A (b-c x))-2 a (A+3 B x)}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2))/x^(5/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 44, normalized size = 0.86 \begin {gather*} \frac {2 \left (-a A-3 a B x-3 A b x+3 A c x^2+3 b B x^2+B c x^3\right )}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2))/x^(5/2),x]

[Out]

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

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fricas [A]  time = 0.42, size = 38, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (B c x^{3} + 3 \, {\left (B b + A c\right )} x^{2} - A a - 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^(5/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 41, normalized size = 0.80 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, B b \sqrt {x} + 2 \, A c \sqrt {x} - \frac {2 \, {\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^(5/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 41, normalized size = 0.80 \begin {gather*} -\frac {2 \left (-B c \,x^{3}-3 A c \,x^{2}-3 B b \,x^{2}+3 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)/x^(5/2),x)

[Out]

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

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maxima [A]  time = 0.66, size = 39, normalized size = 0.76 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, {\left (B b + A c\right )} \sqrt {x} - \frac {2 \, {\left (A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 41, normalized size = 0.80 \begin {gather*} -\frac {2\,A\,a+6\,A\,b\,x+6\,B\,a\,x-6\,A\,c\,x^2-6\,B\,b\,x^2-2\,B\,c\,x^3}{3\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2))/x^(5/2),x)

[Out]

-(2*A*a + 6*A*b*x + 6*B*a*x - 6*A*c*x^2 - 6*B*b*x^2 - 2*B*c*x^3)/(3*x^(3/2))

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sympy [A]  time = 0.81, size = 63, normalized size = 1.24 \begin {gather*} - \frac {2 A a}{3 x^{\frac {3}{2}}} - \frac {2 A b}{\sqrt {x}} + 2 A c \sqrt {x} - \frac {2 B a}{\sqrt {x}} + 2 B b \sqrt {x} + \frac {2 B c x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)/x**(5/2),x)

[Out]

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

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