3.2.44 \(\int \frac {(A+B x) (b x+c x^2)}{\sqrt {x}} \, dx\)

Optimal. Leaf size=39 \[ \frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{3} A b x^{3/2}+\frac {2}{7} B c x^{7/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, 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}{5} x^{5/2} (A c+b B)+\frac {2}{3} A b x^{3/2}+\frac {2}{7} B c x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b*x^(3/2))/3 + (2*(b*B + A*c)*x^(5/2))/5 + (2*B*c*x^(7/2))/7

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 )}{\sqrt {x}} \, dx &=\int \left (A b \sqrt {x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=\frac {2}{3} A b x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{105} x^{3/2} (7 A (5 b+3 c x)+3 B x (7 b+5 c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(7*A*(5*b + 3*c*x) + 3*B*x*(7*b + 5*c*x)))/105

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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 1.05 \begin {gather*} \frac {2}{105} \left (35 A b x^{3/2}+21 A c x^{5/2}+21 b B x^{5/2}+15 B c x^{7/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(35*A*b*x^(3/2) + 21*b*B*x^(5/2) + 21*A*c*x^(5/2) + 15*B*c*x^(7/2)))/105

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fricas [A]  time = 0.39, size = 30, normalized size = 0.77 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c x^{3} + 35 \, A b x + 21 \, {\left (B b + A c\right )} x^{2}\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^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*B*c*x^3 + 35*A*b*x + 21*(B*b + A*c)*x^2)*sqrt(x)

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giac [A]  time = 0.15, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, A b 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^(1/2),x, algorithm="giac")

[Out]

2/7*B*c*x^(7/2) + 2/5*B*b*x^(5/2) + 2/5*A*c*x^(5/2) + 2/3*A*b*x^(3/2)

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maple [A]  time = 0.05, size = 28, normalized size = 0.72 \begin {gather*} \frac {2 \left (15 B c \,x^{2}+21 A c x +21 B b x +35 A b \right ) x^{\frac {3}{2}}}{105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^(1/2),x)

[Out]

2/105*x^(3/2)*(15*B*c*x^2+21*A*c*x+21*B*b*x+35*A*b)

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maxima [A]  time = 0.50, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^(1/2),x, algorithm="maxima")

[Out]

2/7*B*c*x^(7/2) + 2/3*A*b*x^(3/2) + 2/5*(B*b + A*c)*x^(5/2)

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mupad [B]  time = 0.04, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{3/2}\,\left (35\,A\,b+21\,A\,c\,x+21\,B\,b\,x+15\,B\,c\,x^2\right )}{105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^(1/2),x)

[Out]

(2*x^(3/2)*(35*A*b + 21*A*c*x + 21*B*b*x + 15*B*c*x^2))/105

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sympy [A]  time = 0.44, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B b x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**(1/2),x)

[Out]

2*A*b*x**(3/2)/3 + 2*A*c*x**(5/2)/5 + 2*B*b*x**(5/2)/5 + 2*B*c*x**(7/2)/7

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