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

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

[Out]

(-2*a^2*A)/(3*x^(3/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + 2*(2*a*b*B + A*(b^2 + 2*a*c))*Sqrt[x] + (2*(b^2*B + 2*A
*b*c + 2*a*B*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 = 0.0552873, 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} \[ -\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{3} x^{3/2} \left (2 a B c+2 A b c+b^2 B\right )+2 \sqrt{x} \left (A \left (2 a c+b^2\right )+2 a b B\right )-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{5} c x^{5/2} (A c+2 b B)+\frac{2}{7} B c^2 x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*A)/(3*x^(3/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + 2*(2*a*b*B + A*(b^2 + 2*a*c))*Sqrt[x] + (2*(b^2*B + 2*A
*b*c + 2*a*B*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 (a+b x+c x^2\right )^2}{x^{5/2}} \, dx &=\int \left (\frac{a^2 A}{x^{5/2}}+\frac{a (2 A b+a B)}{x^{3/2}}+\frac{2 a b B+A \left (b^2+2 a c\right )}{\sqrt{x}}+\left (b^2 B+2 A b c+2 a B c\right ) \sqrt{x}+c (2 b B+A c) x^{3/2}+B c^2 x^{5/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a (2 A b+a B)}{\sqrt{x}}+2 \left (2 a b B+A \left (b^2+2 a c\right )\right ) \sqrt{x}+\frac{2}{3} \left (b^2 B+2 A b c+2 a B c\right ) 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 = 0.123884, size = 94, normalized size = 0.86 \[ \frac{2 \left (-35 a^2 (A+3 B x)+70 a x (B x (3 b+c x)-3 A (b-c x))+x^2 \left (7 A \left (15 b^2+10 b c x+3 c^2 x^2\right )+B x \left (35 b^2+42 b c x+15 c^2 x^2\right )\right )\right )}{105 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*a^2*(A + 3*B*x) + 70*a*x*(-3*A*(b - c*x) + B*x*(3*b + c*x)) + x^2*(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*x^(3/2))

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Maple [A]  time = 0.006, size = 102, normalized size = 0.9 \begin{align*} -{\frac{-30\,B{c}^{2}{x}^{5}-42\,A{c}^{2}{x}^{4}-84\,B{x}^{4}bc-140\,A{x}^{3}bc-140\,aBc{x}^{3}-70\,{b}^{2}B{x}^{3}-420\,aAc{x}^{2}-210\,A{b}^{2}{x}^{2}-420\,B{x}^{2}ab+420\,aAbx+210\,{a}^{2}Bx+70\,A{a}^{2}}{105}{x}^{-{\frac{3}{2}}}} \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^(5/2),x)

[Out]

-2/105*(-15*B*c^2*x^5-21*A*c^2*x^4-42*B*b*c*x^4-70*A*b*c*x^3-70*B*a*c*x^3-35*B*b^2*x^3-210*A*a*c*x^2-105*A*b^2
*x^2-210*B*a*b*x^2+210*A*a*b*x+105*B*a^2*x+35*A*a^2)/x^(3/2)

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Maxima [A]  time = 1.04966, size = 126, normalized size = 1.16 \begin{align*} \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{3}{2}} + 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} \sqrt{x} - \frac{2 \,{\left (A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, 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+a)^2/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.0222, size = 225, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (15 \, B c^{2} x^{5} + 21 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 35 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 35 \, A a^{2} + 105 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 105 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, 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+a)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*B*c^2*x^5 + 21*(2*B*b*c + A*c^2)*x^4 + 35*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 35*A*a^2 + 105*(2*B*a*b +
A*b^2 + 2*A*a*c)*x^2 - 105*(B*a^2 + 2*A*a*b)*x)/x^(3/2)

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Sympy [A]  time = 4.26485, size = 153, normalized size = 1.4 \begin{align*} - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 A a b}{\sqrt{x}} + 4 A a c \sqrt{x} + 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 a^{2}}{\sqrt{x}} + 4 B a b \sqrt{x} + \frac{4 B a c x^{\frac{3}{2}}}{3} + \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+a)**2/x**(5/2),x)

[Out]

-2*A*a**2/(3*x**(3/2)) - 4*A*a*b/sqrt(x) + 4*A*a*c*sqrt(x) + 2*A*b**2*sqrt(x) + 4*A*b*c*x**(3/2)/3 + 2*A*c**2*
x**(5/2)/5 - 2*B*a**2/sqrt(x) + 4*B*a*b*sqrt(x) + 4*B*a*c*x**(3/2)/3 + 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 = 1.15676, size = 136, normalized size = 1.25 \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} \, B a c x^{\frac{3}{2}} + \frac{4}{3} \, A b c x^{\frac{3}{2}} + 4 \, B a b \sqrt{x} + 2 \, A b^{2} \sqrt{x} + 4 \, A a c \sqrt{x} - \frac{2 \,{\left (3 \, B a^{2} x + 6 \, A a b x + A a^{2}\right )}}{3 \, 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+a)^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*B*a*c*x^(3/2) + 4/3*A*b*c*
x^(3/2) + 4*B*a*b*sqrt(x) + 2*A*b^2*sqrt(x) + 4*A*a*c*sqrt(x) - 2/3*(3*B*a^2*x + 6*A*a*b*x + A*a^2)/x^(3/2)