3.10.11 \(\int \frac {(A+B x) (a+b x+c x^2)}{x^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ 2 \sqrt {x} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{3} x^{3/2} (A c+b B)+\frac {2}{5} B c x^{5/2} \]

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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*} 2 \sqrt {x} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{3} x^{3/2} (A c+b B)+\frac {2}{5} B c x^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*A)/Sqrt[x] + 2*(A*b + a*B)*Sqrt[x] + (2*(b*B + A*c)*x^(3/2))/3 + (2*B*c*x^(5/2))/5

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

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Mathematica [A]  time = 0.04, size = 44, normalized size = 0.86 \begin {gather*} \frac {2 x (5 A (3 b+c x)+B x (5 b+3 c x))-30 a (A-B x)}{15 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*a*(A - B*x) + 2*x*(5*A*(3*b + c*x) + B*x*(5*b + 3*c*x)))/(15*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.03, size = 45, normalized size = 0.88 \begin {gather*} \frac {2 \left (-15 a A+15 a B x+15 A b x+5 A c x^2+5 b B x^2+3 B c x^3\right )}{15 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 39, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (3 \, B c x^{3} + 5 \, {\left (B b + A c\right )} x^{2} - 15 \, A a + 15 \, {\left (B a + A b\right )} x\right )}}{15 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 43, normalized size = 0.84 \begin {gather*} \frac {2}{5} \, B c x^{\frac {5}{2}} + \frac {2}{3} \, B b x^{\frac {3}{2}} + \frac {2}{3} \, A c x^{\frac {3}{2}} + 2 \, B a \sqrt {x} + 2 \, A b \sqrt {x} - \frac {2 \, A a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 42, normalized size = 0.82 \begin {gather*} -\frac {2 \left (-3 B c \,x^{3}-5 A c \,x^{2}-5 B b \,x^{2}-15 A b x -15 B a x +15 A a \right )}{15 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-3*B*c*x^3-5*A*c*x^2-5*B*b*x^2-15*A*b*x-15*B*a*x+15*A*a)/x^(1/2)

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maxima [A]  time = 0.56, size = 39, normalized size = 0.76 \begin {gather*} \frac {2}{5} \, B c x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B b + A c\right )} x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} + 2 \, {\left (B a + A b\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+a)/x^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 41, normalized size = 0.80 \begin {gather*} \sqrt {x}\,\left (2\,A\,b+2\,B\,a\right )+x^{3/2}\,\left (\frac {2\,A\,c}{3}+\frac {2\,B\,b}{3}\right )-\frac {2\,A\,a}{\sqrt {x}}+\frac {2\,B\,c\,x^{5/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*(2*A*b + 2*B*a) + x^(3/2)*((2*A*c)/3 + (2*B*b)/3) - (2*A*a)/x^(1/2) + (2*B*c*x^(5/2))/5

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sympy [A]  time = 0.64, size = 65, normalized size = 1.27 \begin {gather*} - \frac {2 A a}{\sqrt {x}} + 2 A b \sqrt {x} + \frac {2 A c x^{\frac {3}{2}}}{3} + 2 B a \sqrt {x} + \frac {2 B b x^{\frac {3}{2}}}{3} + \frac {2 B c x^{\frac {5}{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)/x**(3/2),x)

[Out]

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

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