3.997 \(\int \frac{(A+B x) (a+b x+c x^2)^2}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0555632, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}+2 \sqrt{x} \left (2 a B c+2 A b c+b^2 B\right )-\frac{2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{\sqrt{x}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+\frac{2}{5} B c^2 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^(7/2),x]

[Out]

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

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

Mathematica [A]  time = 0.116475, size = 95, normalized size = 0.87 \[ \frac{-2 a^2 (3 A+5 B x)-20 a x (A (b+3 c x)+3 B x (b-c x))+2 x^2 \left (5 A \left (-3 b^2+6 b c x+c^2 x^2\right )+B x \left (15 b^2+10 b c x+3 c^2 x^2\right )\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^(7/2),x]

[Out]

(-2*a^2*(3*A + 5*B*x) - 20*a*x*(3*B*x*(b - c*x) + A*(b + 3*c*x)) + 2*x^2*(5*A*(-3*b^2 + 6*b*c*x + c^2*x^2) + B
*x*(15*b^2 + 10*b*c*x + 3*c^2*x^2)))/(15*x^(5/2))

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Maple [A]  time = 0.006, size = 102, normalized size = 0.9 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{5}-10\,A{c}^{2}{x}^{4}-20\,B{x}^{4}bc-60\,A{x}^{3}bc-60\,aBc{x}^{3}-30\,{b}^{2}B{x}^{3}+60\,aAc{x}^{2}+30\,A{b}^{2}{x}^{2}+60\,B{x}^{2}ab+20\,aAbx+10\,{a}^{2}Bx+6\,A{a}^{2}}{15}{x}^{-{\frac{5}{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^(7/2),x)

[Out]

-2/15*(-3*B*c^2*x^5-5*A*c^2*x^4-10*B*b*c*x^4-30*A*b*c*x^3-30*B*a*c*x^3-15*B*b^2*x^3+30*A*a*c*x^2+15*A*b^2*x^2+
30*B*a*b*x^2+10*A*a*b*x+5*B*a^2*x+3*A*a^2)/x^(5/2)

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Maxima [A]  time = 1.15349, size = 127, normalized size = 1.17 \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac{5}{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^(7/2),x, algorithm="maxima")

[Out]

2/5*B*c^2*x^(5/2) + 2/3*(2*B*b*c + A*c^2)*x^(3/2) + 2*(B*b^2 + 2*(B*a + A*b)*c)*sqrt(x) - 2/15*(3*A*a^2 + 15*(
2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 5*(B*a^2 + 2*A*a*b)*x)/x^(5/2)

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Fricas [A]  time = 1.03173, size = 216, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{5} + 5 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 3 \, A a^{2} - 15 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 5 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac{5}{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^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.10547, size = 151, normalized size = 1.39 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{3 x^{\frac{3}{2}}} - \frac{4 A a c}{\sqrt{x}} - \frac{2 A b^{2}}{\sqrt{x}} + 4 A b c \sqrt{x} + \frac{2 A c^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 B a b}{\sqrt{x}} + 4 B a c \sqrt{x} + 2 B b^{2} \sqrt{x} + \frac{4 B b c x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} x^{\frac{5}{2}}}{5} \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**(7/2),x)

[Out]

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

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Giac [A]  time = 1.1621, size = 138, normalized size = 1.27 \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B b c x^{\frac{3}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 2 \, B b^{2} \sqrt{x} + 4 \, B a c \sqrt{x} + 4 \, A b c \sqrt{x} - \frac{2 \,{\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 30 \, A a c x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{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^(7/2),x, algorithm="giac")

[Out]

2/5*B*c^2*x^(5/2) + 4/3*B*b*c*x^(3/2) + 2/3*A*c^2*x^(3/2) + 2*B*b^2*sqrt(x) + 4*B*a*c*sqrt(x) + 4*A*b*c*sqrt(x
) - 2/15*(30*B*a*b*x^2 + 15*A*b^2*x^2 + 30*A*a*c*x^2 + 5*B*a^2*x + 10*A*a*b*x + 3*A*a^2)/x^(5/2)