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

Optimal. Leaf size=43 \[ 2 a A \sqrt{x}+\frac{2}{3} a B x^{3/2}+\frac{2}{5} A c x^{5/2}+\frac{2}{7} B c x^{7/2} \]

[Out]

2*a*A*Sqrt[x] + (2*a*B*x^(3/2))/3 + (2*A*c*x^(5/2))/5 + (2*B*c*x^(7/2))/7

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Rubi [A]  time = 0.0121786, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ 2 a A \sqrt{x}+\frac{2}{3} a B x^{3/2}+\frac{2}{5} A c x^{5/2}+\frac{2}{7} B c x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*A*Sqrt[x] + (2*a*B*x^(3/2))/3 + (2*A*c*x^(5/2))/5 + (2*B*c*x^(7/2))/7

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0127366, size = 34, normalized size = 0.79 \[ \frac{2}{105} \sqrt{x} \left (35 a (3 A+B x)+3 c x^2 (7 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(35*a*(3*A + B*x) + 3*c*x^2*(7*A + 5*B*x)))/105

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Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*}{\frac{30\,Bc{x}^{3}+42\,Ac{x}^{2}+70\,aBx+210\,aA}{105}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*x^(1/2)*(15*B*c*x^3+21*A*c*x^2+35*B*a*x+105*A*a)

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Maxima [A]  time = 0.983477, size = 39, normalized size = 0.91 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.27349, size = 84, normalized size = 1.95 \begin{align*} \frac{2}{105} \,{\left (15 \, B c x^{3} + 21 \, A c x^{2} + 35 \, B a x + 105 \, A a\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.695626, size = 44, normalized size = 1.02 \begin{align*} 2 A a \sqrt{x} + \frac{2 A c x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16405, size = 39, normalized size = 0.91 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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