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

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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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 {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 0.79 \begin {gather*} \frac {2}{105} \sqrt {x} \left (35 a (3 A+B x)+3 c x^2 (7 A+5 B x)\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 0.95 \begin {gather*} \frac {2}{105} \left (105 a A \sqrt {x}+35 a B x^{3/2}+21 A c x^{5/2}+15 B c x^{7/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(105*a*A*Sqrt[x] + 35*a*B*x^(3/2) + 21*A*c*x^(5/2) + 15*B*c*x^(7/2)))/105

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fricas [A]  time = 0.41, size = 29, normalized size = 0.67 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 29, normalized size = 0.67 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.04, size = 30, normalized size = 0.70 \begin {gather*} \frac {2 \left (15 B c \,x^{3}+21 A c \,x^{2}+35 B a x +105 a A \right ) \sqrt {x}}{105} \end {gather*}

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 = 0.49, size = 29, normalized size = 0.67 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 29, normalized size = 0.67 \begin {gather*} 2\,A\,a\,\sqrt {x}+\frac {2\,B\,a\,x^{3/2}}{3}+\frac {2\,A\,c\,x^{5/2}}{5}+\frac {2\,B\,c\,x^{7/2}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x^(1/2) + (2*B*a*x^(3/2))/3 + (2*A*c*x^(5/2))/5 + (2*B*c*x^(7/2))/7

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sympy [A]  time = 0.45, size = 44, normalized size = 1.02 \begin {gather*} 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 {gather*}

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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