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

Optimal. Leaf size=53 \[ \frac {2}{3} x^{3/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{7} B c x^{7/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 53, 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*} \frac {2}{3} x^{3/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{7} B c x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*A*Sqrt[x] + (2*(A*b + 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 (a+b x+c x^2\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a A}{\sqrt {x}}+(A b+a B) \sqrt {x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=2 a A \sqrt {x}+\frac {2}{3} (A b+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 = 0.05, size = 46, normalized size = 0.87 \begin {gather*} \frac {2}{105} \sqrt {x} (35 a (3 A+B x)+x (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)*(a + b*x + c*x^2))/Sqrt[x],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 59, normalized size = 1.11 \begin {gather*} \frac {2}{105} \left (105 a A \sqrt {x}+35 a B x^{3/2}+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)*(a + b*x + c*x^2))/Sqrt[x],x]

[Out]

(2*(105*a*A*Sqrt[x] + 35*A*b*x^(3/2) + 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 = 0.42, size = 39, normalized size = 0.74 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c x^{3} + 21 \, {\left (B b + A c\right )} x^{2} + 105 \, A a + 35 \, {\left (B a + A b\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+a)/x^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 43, normalized size = 0.81 \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} \, B a x^{\frac {3}{2}} + \frac {2}{3} \, A b 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+b*x+a)/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*B*a*x^(3/2) + 2/3*A*b*x^(3/2) + 2*A*a*sqrt(x)

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maple [A]  time = 0.05, size = 42, normalized size = 0.79 \begin {gather*} \frac {2 \left (15 B c \,x^{3}+21 A c \,x^{2}+21 B b \,x^{2}+35 A b x +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+b*x+a)/x^(1/2),x)

[Out]

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

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maxima [A]  time = 0.50, size = 39, normalized size = 0.74 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} + 2 \, A a \sqrt {x} + \frac {2}{3} \, {\left (B a + A b\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+a)/x^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 41, normalized size = 0.77 \begin {gather*} x^{3/2}\,\left (\frac {2\,A\,b}{3}+\frac {2\,B\,a}{3}\right )+x^{5/2}\,\left (\frac {2\,A\,c}{5}+\frac {2\,B\,b}{5}\right )+2\,A\,a\,\sqrt {x}+\frac {2\,B\,c\,x^{7/2}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.47, size = 68, normalized size = 1.28 \begin {gather*} 2 A a \sqrt {x} + \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B a x^{\frac {3}{2}}}{3} + \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+a)/x**(1/2),x)

[Out]

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

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