3.755 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{9/2}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{10 a^3 b^2 (3 a B+4 A b)}{\sqrt{x}}+10 a^2 b^3 \sqrt{x} (4 a B+3 A b)-\frac{2 a^5 (a B+6 A b)}{5 x^{5/2}}-\frac{2 a^4 b (2 a B+5 A b)}{x^{3/2}}-\frac{2 a^6 A}{7 x^{7/2}}+2 a b^4 x^{3/2} (5 a B+2 A b)+\frac{2}{5} b^5 x^{5/2} (6 a B+A b)+\frac{2}{7} b^6 B x^{7/2} \]

[Out]

(-2*a^6*A)/(7*x^(7/2)) - (2*a^5*(6*A*b + a*B))/(5*x^(5/2)) - (2*a^4*b*(5*A*b + 2*a*B))/x^(3/2) - (10*a^3*b^2*(
4*A*b + 3*a*B))/Sqrt[x] + 10*a^2*b^3*(3*A*b + 4*a*B)*Sqrt[x] + 2*a*b^4*(2*A*b + 5*a*B)*x^(3/2) + (2*b^5*(A*b +
6*a*B)*x^(5/2))/5 + (2*b^6*B*x^(7/2))/7

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Rubi [A]  time = 0.0819405, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 76} \[ -\frac{10 a^3 b^2 (3 a B+4 A b)}{\sqrt{x}}+10 a^2 b^3 \sqrt{x} (4 a B+3 A b)-\frac{2 a^5 (a B+6 A b)}{5 x^{5/2}}-\frac{2 a^4 b (2 a B+5 A b)}{x^{3/2}}-\frac{2 a^6 A}{7 x^{7/2}}+2 a b^4 x^{3/2} (5 a B+2 A b)+\frac{2}{5} b^5 x^{5/2} (6 a B+A b)+\frac{2}{7} b^6 B x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(9/2),x]

[Out]

(-2*a^6*A)/(7*x^(7/2)) - (2*a^5*(6*A*b + a*B))/(5*x^(5/2)) - (2*a^4*b*(5*A*b + 2*a*B))/x^(3/2) - (10*a^3*b^2*(
4*A*b + 3*a*B))/Sqrt[x] + 10*a^2*b^3*(3*A*b + 4*a*B)*Sqrt[x] + 2*a*b^4*(2*A*b + 5*a*B)*x^(3/2) + (2*b^5*(A*b +
6*a*B)*x^(5/2))/5 + (2*b^6*B*x^(7/2))/7

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{9/2}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^{9/2}} \, dx\\ &=\int \left (\frac{a^6 A}{x^{9/2}}+\frac{a^5 (6 A b+a B)}{x^{7/2}}+\frac{3 a^4 b (5 A b+2 a B)}{x^{5/2}}+\frac{5 a^3 b^2 (4 A b+3 a B)}{x^{3/2}}+\frac{5 a^2 b^3 (3 A b+4 a B)}{\sqrt{x}}+3 a b^4 (2 A b+5 a B) \sqrt{x}+b^5 (A b+6 a B) x^{3/2}+b^6 B x^{5/2}\right ) \, dx\\ &=-\frac{2 a^6 A}{7 x^{7/2}}-\frac{2 a^5 (6 A b+a B)}{5 x^{5/2}}-\frac{2 a^4 b (5 A b+2 a B)}{x^{3/2}}-\frac{10 a^3 b^2 (4 A b+3 a B)}{\sqrt{x}}+10 a^2 b^3 (3 A b+4 a B) \sqrt{x}+2 a b^4 (2 A b+5 a B) x^{3/2}+\frac{2}{5} b^5 (A b+6 a B) x^{5/2}+\frac{2}{7} b^6 B x^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0393571, size = 123, normalized size = 0.81 \[ \frac{2 \left (-175 a^4 b^2 x^2 (A+3 B x)+700 a^3 b^3 x^3 (B x-A)+175 a^2 b^4 x^4 (3 A+B x)-14 a^5 b x (3 A+5 B x)+a^6 (-(5 A+7 B x))+14 a b^5 x^5 (5 A+3 B x)+b^6 x^6 (7 A+5 B x)\right )}{35 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(9/2),x]

[Out]

(2*(700*a^3*b^3*x^3*(-A + B*x) + 175*a^2*b^4*x^4*(3*A + B*x) - 175*a^4*b^2*x^2*(A + 3*B*x) + 14*a*b^5*x^5*(5*A
+ 3*B*x) - 14*a^5*b*x*(3*A + 5*B*x) + b^6*x^6*(7*A + 5*B*x) - a^6*(5*A + 7*B*x)))/(35*x^(7/2))

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Maple [A]  time = 0.007, size = 148, normalized size = 1. \begin{align*} -{\frac{-10\,B{b}^{6}{x}^{7}-14\,A{b}^{6}{x}^{6}-84\,B{x}^{6}a{b}^{5}-140\,aA{b}^{5}{x}^{5}-350\,B{x}^{5}{a}^{2}{b}^{4}-1050\,{a}^{2}A{b}^{4}{x}^{4}-1400\,B{x}^{4}{a}^{3}{b}^{3}+1400\,{a}^{3}A{b}^{3}{x}^{3}+1050\,B{x}^{3}{a}^{4}{b}^{2}+350\,{a}^{4}A{b}^{2}{x}^{2}+140\,B{x}^{2}{a}^{5}b+84\,{a}^{5}Abx+14\,B{a}^{6}x+10\,A{a}^{6}}{35}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x)

[Out]

-2/35*(-5*B*b^6*x^7-7*A*b^6*x^6-42*B*a*b^5*x^6-70*A*a*b^5*x^5-175*B*a^2*b^4*x^5-525*A*a^2*b^4*x^4-700*B*a^3*b^
3*x^4+700*A*a^3*b^3*x^3+525*B*a^4*b^2*x^3+175*A*a^4*b^2*x^2+70*B*a^5*b*x^2+42*A*a^5*b*x+7*B*a^6*x+5*A*a^6)/x^(
7/2)

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Maxima [A]  time = 1.01077, size = 200, normalized size = 1.32 \begin{align*} \frac{2}{7} \, B b^{6} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{5}{2}} + 2 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{3}{2}} + 10 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \sqrt{x} - \frac{2 \,{\left (5 \, A a^{6} + 175 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 35 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 7 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x, algorithm="maxima")

[Out]

2/7*B*b^6*x^(7/2) + 2/5*(6*B*a*b^5 + A*b^6)*x^(5/2) + 2*(5*B*a^2*b^4 + 2*A*a*b^5)*x^(3/2) + 10*(4*B*a^3*b^3 +
3*A*a^2*b^4)*sqrt(x) - 2/35*(5*A*a^6 + 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 35*(2*B*a^5*b + 5*A*a^4*b^2)*x^2
+ 7*(B*a^6 + 6*A*a^5*b)*x)/x^(7/2)

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Fricas [A]  time = 1.58436, size = 321, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (5 \, B b^{6} x^{7} - 5 \, A a^{6} + 7 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 35 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 175 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 175 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 35 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 7 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x, algorithm="fricas")

[Out]

2/35*(5*B*b^6*x^7 - 5*A*a^6 + 7*(6*B*a*b^5 + A*b^6)*x^6 + 35*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 175*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*x^4 - 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 35*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 7*(B*a^6 + 6*A*a
^5*b)*x)/x^(7/2)

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Sympy [A]  time = 11.3763, size = 202, normalized size = 1.34 \begin{align*} - \frac{2 A a^{6}}{7 x^{\frac{7}{2}}} - \frac{12 A a^{5} b}{5 x^{\frac{5}{2}}} - \frac{10 A a^{4} b^{2}}{x^{\frac{3}{2}}} - \frac{40 A a^{3} b^{3}}{\sqrt{x}} + 30 A a^{2} b^{4} \sqrt{x} + 4 A a b^{5} x^{\frac{3}{2}} + \frac{2 A b^{6} x^{\frac{5}{2}}}{5} - \frac{2 B a^{6}}{5 x^{\frac{5}{2}}} - \frac{4 B a^{5} b}{x^{\frac{3}{2}}} - \frac{30 B a^{4} b^{2}}{\sqrt{x}} + 40 B a^{3} b^{3} \sqrt{x} + 10 B a^{2} b^{4} x^{\frac{3}{2}} + \frac{12 B a b^{5} x^{\frac{5}{2}}}{5} + \frac{2 B b^{6} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**(9/2),x)

[Out]

-2*A*a**6/(7*x**(7/2)) - 12*A*a**5*b/(5*x**(5/2)) - 10*A*a**4*b**2/x**(3/2) - 40*A*a**3*b**3/sqrt(x) + 30*A*a*
*2*b**4*sqrt(x) + 4*A*a*b**5*x**(3/2) + 2*A*b**6*x**(5/2)/5 - 2*B*a**6/(5*x**(5/2)) - 4*B*a**5*b/x**(3/2) - 30
*B*a**4*b**2/sqrt(x) + 40*B*a**3*b**3*sqrt(x) + 10*B*a**2*b**4*x**(3/2) + 12*B*a*b**5*x**(5/2)/5 + 2*B*b**6*x*
*(7/2)/7

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Giac [A]  time = 1.1445, size = 200, normalized size = 1.32 \begin{align*} \frac{2}{7} \, B b^{6} x^{\frac{7}{2}} + \frac{12}{5} \, B a b^{5} x^{\frac{5}{2}} + \frac{2}{5} \, A b^{6} x^{\frac{5}{2}} + 10 \, B a^{2} b^{4} x^{\frac{3}{2}} + 4 \, A a b^{5} x^{\frac{3}{2}} + 40 \, B a^{3} b^{3} \sqrt{x} + 30 \, A a^{2} b^{4} \sqrt{x} - \frac{2 \,{\left (525 \, B a^{4} b^{2} x^{3} + 700 \, A a^{3} b^{3} x^{3} + 70 \, B a^{5} b x^{2} + 175 \, A a^{4} b^{2} x^{2} + 7 \, B a^{6} x + 42 \, A a^{5} b x + 5 \, A a^{6}\right )}}{35 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x, algorithm="giac")

[Out]

2/7*B*b^6*x^(7/2) + 12/5*B*a*b^5*x^(5/2) + 2/5*A*b^6*x^(5/2) + 10*B*a^2*b^4*x^(3/2) + 4*A*a*b^5*x^(3/2) + 40*B
*a^3*b^3*sqrt(x) + 30*A*a^2*b^4*sqrt(x) - 2/35*(525*B*a^4*b^2*x^3 + 700*A*a^3*b^3*x^3 + 70*B*a^5*b*x^2 + 175*A
*a^4*b^2*x^2 + 7*B*a^6*x + 42*A*a^5*b*x + 5*A*a^6)/x^(7/2)