3.998 \(\int \frac{(A+B x) (a+b x+c x^2)^2}{x^{9/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0564041, 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} \[ -\frac{2 a^2 A}{7 x^{7/2}}-\frac{2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{3 x^{3/2}}-\frac{2 \left (2 a B c+2 A b c+b^2 B\right )}{\sqrt{x}}-\frac{2 a (a B+2 A b)}{5 x^{5/2}}+2 c \sqrt{x} (A c+2 b B)+\frac{2}{3} B c^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^(9/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0870226, size = 95, normalized size = 0.87 \[ -\frac{2 \left (3 a^2 (5 A+7 B x)+14 a x (A (3 b+5 c x)+5 B x (b+3 c x))+35 x^2 \left (A \left (b^2+6 b c x-3 c^2 x^2\right )-B x \left (-3 b^2+6 b c x+c^2 x^2\right )\right )\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^(9/2),x]

[Out]

(-2*(3*a^2*(5*A + 7*B*x) + 14*a*x*(5*B*x*(b + 3*c*x) + A*(3*b + 5*c*x)) + 35*x^2*(A*(b^2 + 6*b*c*x - 3*c^2*x^2
) - B*x*(-3*b^2 + 6*b*c*x + c^2*x^2))))/(105*x^(7/2))

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Maple [A]  time = 0.006, size = 102, normalized size = 0.9 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}-210\,A{c}^{2}{x}^{4}-420\,B{x}^{4}bc+420\,A{x}^{3}bc+420\,aBc{x}^{3}+210\,{b}^{2}B{x}^{3}+140\,aAc{x}^{2}+70\,A{b}^{2}{x}^{2}+140\,B{x}^{2}ab+84\,aAbx+42\,{a}^{2}Bx+30\,A{a}^{2}}{105}{x}^{-{\frac{7}{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^(9/2),x)

[Out]

-2/105*(-35*B*c^2*x^5-105*A*c^2*x^4-210*B*b*c*x^4+210*A*b*c*x^3+210*B*a*c*x^3+105*B*b^2*x^3+70*A*a*c*x^2+35*A*
b^2*x^2+70*B*a*b*x^2+42*A*a*b*x+21*B*a^2*x+15*A*a^2)/x^(7/2)

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Maxima [A]  time = 1.10709, size = 127, normalized size = 1.17 \begin{align*} \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 2 \,{\left (2 \, B b c + A c^{2}\right )} \sqrt{x} - \frac{2 \,{\left (105 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 15 \, A a^{2} + 35 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac{7}{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^(9/2),x, algorithm="maxima")

[Out]

2/3*B*c^2*x^(3/2) + 2*(2*B*b*c + A*c^2)*sqrt(x) - 2/105*(105*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 15*A*a^2 + 35*(2*
B*a*b + A*b^2 + 2*A*a*c)*x^2 + 21*(B*a^2 + 2*A*a*b)*x)/x^(7/2)

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Fricas [A]  time = 1.0474, size = 225, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (35 \, B c^{2} x^{5} + 105 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} - 105 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 15 \, A a^{2} - 35 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 21 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac{7}{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^(9/2),x, algorithm="fricas")

[Out]

2/105*(35*B*c^2*x^5 + 105*(2*B*b*c + A*c^2)*x^4 - 105*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 15*A*a^2 - 35*(2*B*a*b +
A*b^2 + 2*A*a*c)*x^2 - 21*(B*a^2 + 2*A*a*b)*x)/x^(7/2)

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Sympy [A]  time = 7.34206, size = 153, normalized size = 1.4 \begin{align*} - \frac{2 A a^{2}}{7 x^{\frac{7}{2}}} - \frac{4 A a b}{5 x^{\frac{5}{2}}} - \frac{4 A a c}{3 x^{\frac{3}{2}}} - \frac{2 A b^{2}}{3 x^{\frac{3}{2}}} - \frac{4 A b c}{\sqrt{x}} + 2 A c^{2} \sqrt{x} - \frac{2 B a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 B a b}{3 x^{\frac{3}{2}}} - \frac{4 B a c}{\sqrt{x}} - \frac{2 B b^{2}}{\sqrt{x}} + 4 B b c \sqrt{x} + \frac{2 B c^{2} x^{\frac{3}{2}}}{3} \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**(9/2),x)

[Out]

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

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Giac [A]  time = 1.18505, size = 138, normalized size = 1.27 \begin{align*} \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 4 \, B b c \sqrt{x} + 2 \, A c^{2} \sqrt{x} - \frac{2 \,{\left (105 \, B b^{2} x^{3} + 210 \, B a c x^{3} + 210 \, A b c x^{3} + 70 \, B a b x^{2} + 35 \, A b^{2} x^{2} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 42 \, A a b x + 15 \, A a^{2}\right )}}{105 \, x^{\frac{7}{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^(9/2),x, algorithm="giac")

[Out]

2/3*B*c^2*x^(3/2) + 4*B*b*c*sqrt(x) + 2*A*c^2*sqrt(x) - 2/105*(105*B*b^2*x^3 + 210*B*a*c*x^3 + 210*A*b*c*x^3 +
70*B*a*b*x^2 + 35*A*b^2*x^2 + 70*A*a*c*x^2 + 21*B*a^2*x + 42*A*a*b*x + 15*A*a^2)/x^(7/2)