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

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{5} b x^{5/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{7} b^2 B x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0273857, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ 2 a^2 A \sqrt{x}+\frac{2}{5} b x^{5/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{7} b^2 B x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0167669, size = 51, normalized size = 0.84 \[ \frac{2}{105} \sqrt{x} \left (35 a^2 (3 A+B x)+14 a b x (5 A+3 B x)+3 b^2 x^2 (7 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(35*a^2*(3*A + B*x) + 14*a*b*x*(5*A + 3*B*x) + 3*b^2*x^2*(7*A + 5*B*x)))/105

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Maple [A]  time = 0.007, size = 52, normalized size = 0.9 \begin{align*}{\frac{30\,{b}^{2}B{x}^{3}+42\,A{b}^{2}{x}^{2}+84\,B{x}^{2}ab+140\,aAbx+70\,{a}^{2}Bx+210\,A{a}^{2}}{105}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*x^(1/2)*(15*B*b^2*x^3+21*A*b^2*x^2+42*B*a*b*x^2+70*A*a*b*x+35*B*a^2*x+105*A*a^2)

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Maxima [A]  time = 1.03928, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48841, size = 127, normalized size = 2.08 \begin{align*} \frac{2}{105} \,{\left (15 \, B b^{2} x^{3} + 105 \, A a^{2} + 21 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.531826, size = 78, normalized size = 1.28 \begin{align*} 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(3/2)/3 + 2*A*b**2*x**(5/2)/5 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(5/2)/5 + 2*B*b
**2*x**(7/2)/7

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Giac [A]  time = 1.31061, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B a b x^{\frac{5}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} + 2 \, A a^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*B*b^2*x^(7/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2) + 2*A*a^2*sq
rt(x)