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

Optimal. Leaf size=37 \[ \frac {2}{3} x^{3/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{5} B c x^{5/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, 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*} \frac {2}{3} x^{3/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{5} B c x^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*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 (b x+c x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac {A b}{\sqrt {x}}+(b B+A c) \sqrt {x}+B c x^{3/2}\right ) \, dx\\ &=2 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 = 0.01, size = 31, normalized size = 0.84 \begin {gather*} \frac {2}{15} \sqrt {x} (5 A (3 b+c x)+B x (5 b+3 c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 1.11 \begin {gather*} \frac {2}{15} \left (15 A b \sqrt {x}+5 A c x^{3/2}+5 b B x^{3/2}+3 B c x^{5/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{15} \, {\left (3 \, B c x^{2} + 15 \, A b + 5 \, {\left (B b + A c\right )} x\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^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 29, normalized size = 0.78 \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 \, 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^(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*A*b*sqrt(x)

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maple [A]  time = 0.04, size = 28, normalized size = 0.76 \begin {gather*} \frac {2 \left (3 B c \,x^{2}+5 A c x +5 B b x +15 A b \right ) \sqrt {x}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{5} \, B c x^{\frac {5}{2}} + 2 \, A b \sqrt {x} + \frac {2}{3} \, {\left (B b + A c\right )} 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)/x^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 27, normalized size = 0.73 \begin {gather*} \frac {2\,\sqrt {x}\,\left (15\,A\,b+5\,A\,c\,x+5\,B\,b\,x+3\,B\,c\,x^2\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.48, size = 44, normalized size = 1.19 \begin {gather*} 2 A b \sqrt {x} + \frac {2 A c x^{\frac {3}{2}}}{3} + \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)/x**(3/2),x)

[Out]

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

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