3.995 \(\int \frac{(A+B x) (a+b x+c x^2)^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{5} x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{3} x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+2 a \sqrt{x} (a B+2 A b)+\frac{2}{7} c x^{7/2} (A c+2 b B)+\frac{2}{9} B c^2 x^{9/2} \]

[Out]

(-2*a^2*A)/Sqrt[x] + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*(2*a*b*B + A*(b^2 + 2*a*c))*x^(3/2))/3 + (2*(b^2*B + 2*A*b
*c + 2*a*B*c)*x^(5/2))/5 + (2*c*(2*b*B + A*c)*x^(7/2))/7 + (2*B*c^2*x^(9/2))/9

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Rubi [A]  time = 0.0606816, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {765} \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{5} x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{3} x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+2 a \sqrt{x} (a B+2 A b)+\frac{2}{7} c x^{7/2} (A c+2 b B)+\frac{2}{9} B c^2 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^(3/2),x]

[Out]

(-2*a^2*A)/Sqrt[x] + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*(2*a*b*B + A*(b^2 + 2*a*c))*x^(3/2))/3 + (2*(b^2*B + 2*A*b
*c + 2*a*B*c)*x^(5/2))/5 + (2*c*(2*b*B + A*c)*x^(7/2))/7 + (2*B*c^2*x^(9/2))/9

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 )^2}{x^{3/2}} \, dx &=\int \left (\frac{a^2 A}{x^{3/2}}+\frac{a (2 A b+a B)}{\sqrt{x}}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) \sqrt{x}+\left (b^2 B+2 A b c+2 a B c\right ) x^{3/2}+c (2 b B+A c) x^{5/2}+B c^2 x^{7/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{\sqrt{x}}+2 a (2 A b+a B) \sqrt{x}+\frac{2}{3} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{3/2}+\frac{2}{5} \left (b^2 B+2 A b c+2 a B c\right ) x^{5/2}+\frac{2}{7} c (2 b B+A c) x^{7/2}+\frac{2}{9} B c^2 x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.119333, size = 97, normalized size = 0.89 \[ \frac{-630 a^2 (A-B x)+84 a x (5 A (3 b+c x)+B x (5 b+3 c x))+2 x^2 \left (3 A \left (35 b^2+42 b c x+15 c^2 x^2\right )+B x \left (63 b^2+90 b c x+35 c^2 x^2\right )\right )}{315 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^(3/2),x]

[Out]

(-630*a^2*(A - B*x) + 84*a*x*(5*A*(3*b + c*x) + B*x*(5*b + 3*c*x)) + 2*x^2*(3*A*(35*b^2 + 42*b*c*x + 15*c^2*x^
2) + B*x*(63*b^2 + 90*b*c*x + 35*c^2*x^2)))/(315*Sqrt[x])

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Maple [A]  time = 0.006, size = 102, normalized size = 0.9 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}-90\,A{c}^{2}{x}^{4}-180\,B{x}^{4}bc-252\,A{x}^{3}bc-252\,aBc{x}^{3}-126\,{b}^{2}B{x}^{3}-420\,aAc{x}^{2}-210\,A{b}^{2}{x}^{2}-420\,B{x}^{2}ab-1260\,aAbx-630\,{a}^{2}Bx+630\,A{a}^{2}}{315}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^2/x^(3/2),x)

[Out]

-2/315*(-35*B*c^2*x^5-45*A*c^2*x^4-90*B*b*c*x^4-126*A*b*c*x^3-126*B*a*c*x^3-63*B*b^2*x^3-210*A*a*c*x^2-105*A*b
^2*x^2-210*B*a*b*x^2-630*A*a*b*x-315*B*a^2*x+315*A*a^2)/x^(1/2)

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Maxima [A]  time = 1.05141, size = 126, normalized size = 1.16 \begin{align*} \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{2} + 2 \, A a b\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+a)^2/x^(3/2),x, algorithm="maxima")

[Out]

2/9*B*c^2*x^(9/2) + 2/7*(2*B*b*c + A*c^2)*x^(7/2) + 2/5*(B*b^2 + 2*(B*a + A*b)*c)*x^(5/2) - 2*A*a^2/sqrt(x) +
2/3*(2*B*a*b + A*b^2 + 2*A*a*c)*x^(3/2) + 2*(B*a^2 + 2*A*a*b)*sqrt(x)

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Fricas [A]  time = 1.06086, size = 227, normalized size = 2.08 \begin{align*} \frac{2 \,{\left (35 \, B c^{2} x^{5} + 45 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 63 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 315 \, A a^{2} + 105 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 315 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{315 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*c^2*x^5 + 45*(2*B*b*c + A*c^2)*x^4 + 63*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 315*A*a^2 + 105*(2*B*a*b +
A*b^2 + 2*A*a*c)*x^2 + 315*(B*a^2 + 2*A*a*b)*x)/sqrt(x)

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Sympy [A]  time = 4.06232, size = 156, normalized size = 1.43 \begin{align*} - \frac{2 A a^{2}}{\sqrt{x}} + 4 A a b \sqrt{x} + \frac{4 A a c x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} + \frac{4 A b c x^{\frac{5}{2}}}{5} + \frac{2 A c^{2} x^{\frac{7}{2}}}{7} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{4 B a c x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{5}{2}}}{5} + \frac{4 B b c x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**2/x**(3/2),x)

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*sqrt(x) + 4*A*a*c*x**(3/2)/3 + 2*A*b**2*x**(3/2)/3 + 4*A*b*c*x**(5/2)/5 + 2*A*c**2
*x**(7/2)/7 + 2*B*a**2*sqrt(x) + 4*B*a*b*x**(3/2)/3 + 4*B*a*c*x**(5/2)/5 + 2*B*b**2*x**(5/2)/5 + 4*B*b*c*x**(7
/2)/7 + 2*B*c**2*x**(9/2)/9

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Giac [A]  time = 1.15933, size = 139, normalized size = 1.28 \begin{align*} \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B b c x^{\frac{7}{2}} + \frac{2}{7} \, A c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} + \frac{4}{5} \, B a c x^{\frac{5}{2}} + \frac{4}{5} \, A b c x^{\frac{5}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a c x^{\frac{3}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{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+a)^2/x^(3/2),x, algorithm="giac")

[Out]

2/9*B*c^2*x^(9/2) + 4/7*B*b*c*x^(7/2) + 2/7*A*c^2*x^(7/2) + 2/5*B*b^2*x^(5/2) + 4/5*B*a*c*x^(5/2) + 4/5*A*b*c*
x^(5/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) + 4/3*A*a*c*x^(3/2) + 2*B*a^2*sqrt(x) + 4*A*a*b*sqrt(x) - 2*A*
a^2/sqrt(x)