3.155 \(\int \frac{(A+B x) (b x+c x^2)^2}{x^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.030016, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ 2 A b^2 \sqrt{x}+\frac{2}{5} c x^{5/2} (A c+2 b B)+\frac{2}{3} b x^{3/2} (2 A c+b B)+\frac{2}{7} B c^2 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0166796, size = 54, normalized size = 0.89 \[ \frac{2}{105} \sqrt{x} \left (7 A \left (15 b^2+10 b c x+3 c^2 x^2\right )+B x \left (35 b^2+42 b c x+15 c^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^(5/2),x]

[Out]

(2*Sqrt[x]*(7*A*(15*b^2 + 10*b*c*x + 3*c^2*x^2) + B*x*(35*b^2 + 42*b*c*x + 15*c^2*x^2)))/105

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Maple [A]  time = 0.005, size = 52, normalized size = 0.9 \begin{align*}{\frac{30\,B{c}^{2}{x}^{3}+42\,A{c}^{2}{x}^{2}+84\,B{x}^{2}bc+140\,Abcx+70\,{b}^{2}Bx+210\,A{b}^{2}}{105}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.00272, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + 2 \, A b^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.78917, size = 127, normalized size = 2.08 \begin{align*} \frac{2}{105} \,{\left (15 \, B c^{2} x^{3} + 105 \, A b^{2} + 21 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 35 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.34169, size = 78, normalized size = 1.28 \begin{align*} 2 A b^{2} \sqrt{x} + \frac{4 A b c x^{\frac{3}{2}}}{3} + \frac{2 A c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3} + \frac{4 B b c x^{\frac{5}{2}}}{5} + \frac{2 B c^{2} 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+b*x)**2/x**(5/2),x)

[Out]

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

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Giac [A]  time = 1.12841, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B b c x^{\frac{5}{2}} + \frac{2}{5} \, A c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A b c x^{\frac{3}{2}} + 2 \, A b^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(5/2),x, algorithm="giac")

[Out]

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