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

Optimal. Leaf size=35 \[ 2 \sqrt {x} (A c+b B)-\frac {2 A b}{\sqrt {x}}+\frac {2}{3} B c x^{3/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} 2 \sqrt {x} (A c+b B)-\frac {2 A b}{\sqrt {x}}+\frac {2}{3} B c x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*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 (b x+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac {A b}{x^{3/2}}+\frac {b B+A c}{\sqrt {x}}+B c \sqrt {x}\right ) \, dx\\ &=-\frac {2 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 = 0.01, size = 29, normalized size = 0.83 \begin {gather*} \frac {2 (B x (3 b+c x)-3 A (b-c x))}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 30, normalized size = 0.86 \begin {gather*} \frac {2 \left (-3 A b+3 A c x+3 b B x+B c x^2\right )}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 26, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (B c x^{2} - 3 \, A b + 3 \, {\left (B b + A c\right )} x\right )}}{3 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 29, normalized size = 0.83 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, B b \sqrt {x} + 2 \, A c \sqrt {x} - \frac {2 \, A b}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/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*A*b/sqrt(x)

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maple [A]  time = 0.05, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2 \left (-B c \,x^{2}-3 A c x -3 B b x +3 A b \right )}{3 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/x^(1/2)*(-B*c*x^2-3*A*c*x-3*B*b*x+3*A*b)

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maxima [A]  time = 0.71, size = 27, normalized size = 0.77 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} - \frac {2 \, A b}{\sqrt {x}} + 2 \, {\left (B b + A c\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.02, size = 27, normalized size = 0.77 \begin {gather*} \frac {6\,A\,c\,x-6\,A\,b+6\,B\,b\,x+2\,B\,c\,x^2}{3\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*A*c*x - 6*A*b + 6*B*b*x + 2*B*c*x^2)/(3*x^(1/2))

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sympy [A]  time = 0.76, size = 41, normalized size = 1.17 \begin {gather*} - \frac {2 A b}{\sqrt {x}} + 2 A c \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)/x**(5/2),x)

[Out]

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

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