\(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^{7/2}} \, dx\) [745]

Optimal result

Integrand size = 29, antiderivative size = 107 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=-\frac {2 a^4 A}{5 x^{5/2}}-\frac {2 a^3 (4 A b+a B)}{3 x^{3/2}}-\frac {4 a^2 b (3 A b+2 a B)}{\sqrt {x}}+4 a b^2 (2 A b+3 a B) \sqrt {x}+\frac {2}{3} b^3 (A b+4 a B) x^{3/2}+\frac {2}{5} b^4 B x^{5/2} \]

[Out]

-2/5*a^4*A/x^(5/2)-2/3*a^3*(4*A*b+B*a)/x^(3/2)+2/3*b^3*(A*b+4*B*a)*x^(3/2)+2/5*b^4*B*x^(5/2)-4*a^2*b*(3*A*b+2*
B*a)/x^(1/2)+4*a*b^2*(2*A*b+3*B*a)*x^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=-\frac {2 a^4 A}{5 x^{5/2}}-\frac {2 a^3 (a B+4 A b)}{3 x^{3/2}}-\frac {4 a^2 b (2 a B+3 A b)}{\sqrt {x}}+\frac {2}{3} b^3 x^{3/2} (4 a B+A b)+4 a b^2 \sqrt {x} (3 a B+2 A b)+\frac {2}{5} b^4 B x^{5/2} \]

[In]

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

[Out]

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

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 77

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2 \left (90 a^2 b^2 x^2 (-A+B x)+20 a b^3 x^3 (3 A+B x)-20 a^3 b x (A+3 B x)+b^4 x^4 (5 A+3 B x)-a^4 (3 A+5 B x)\right )}{15 x^{5/2}} \]

[In]

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

[Out]

(2*(90*a^2*b^2*x^2*(-A + B*x) + 20*a*b^3*x^3*(3*A + B*x) - 20*a^3*b*x*(A + 3*B*x) + b^4*x^4*(5*A + 3*B*x) - a^
4*(3*A + 5*B*x)))/(15*x^(5/2))

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {2 b^{4} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{4} x^{\frac {3}{2}}}{3}+\frac {8 B a \,b^{3} x^{\frac {3}{2}}}{3}+8 A a \,b^{3} \sqrt {x}+12 B \,a^{2} b^{2} \sqrt {x}-\frac {2 a^{4} A}{5 x^{\frac {5}{2}}}-\frac {2 a^{3} \left (4 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {4 a^{2} b \left (3 A b +2 B a \right )}{\sqrt {x}}\)	\(95\)
default	\(\frac {2 b^{4} B \,x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{4} x^{\frac {3}{2}}}{3}+\frac {8 B a \,b^{3} x^{\frac {3}{2}}}{3}+8 A a \,b^{3} \sqrt {x}+12 B \,a^{2} b^{2} \sqrt {x}-\frac {2 a^{4} A}{5 x^{\frac {5}{2}}}-\frac {2 a^{3} \left (4 A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {4 a^{2} b \left (3 A b +2 B a \right )}{\sqrt {x}}\)	\(95\)
gosper	\(-\frac {2 \left (-3 b^{4} B \,x^{5}-5 A \,b^{4} x^{4}-20 x^{4} B \,b^{3} a -60 a A \,b^{3} x^{3}-90 x^{3} B \,a^{2} b^{2}+90 a^{2} A \,b^{2} x^{2}+60 x^{2} B \,a^{3} b +20 a^{3} A b x +5 a^{4} B x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}}}\)	\(100\)
trager	\(-\frac {2 \left (-3 b^{4} B \,x^{5}-5 A \,b^{4} x^{4}-20 x^{4} B \,b^{3} a -60 a A \,b^{3} x^{3}-90 x^{3} B \,a^{2} b^{2}+90 a^{2} A \,b^{2} x^{2}+60 x^{2} B \,a^{3} b +20 a^{3} A b x +5 a^{4} B x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}}}\)	\(100\)
risch	\(-\frac {2 \left (-3 b^{4} B \,x^{5}-5 A \,b^{4} x^{4}-20 x^{4} B \,b^{3} a -60 a A \,b^{3} x^{3}-90 x^{3} B \,a^{2} b^{2}+90 a^{2} A \,b^{2} x^{2}+60 x^{2} B \,a^{3} b +20 a^{3} A b x +5 a^{4} B x +3 A \,a^{4}\right )}{15 x^{\frac {5}{2}}}\)	\(100\)

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/5*b^4*B*x^(5/2)+2/3*A*b^4*x^(3/2)+8/3*B*a*b^3*x^(3/2)+8*A*a*b^3*x^(1/2)+12*B*a^2*b^2*x^(1/2)-2/5*a^4*A/x^(5/
2)-2/3*a^3*(4*A*b+B*a)/x^(3/2)-4*a^2*b*(3*A*b+2*B*a)/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2 \, {\left (3 \, B b^{4} x^{5} - 3 \, A a^{4} + 5 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 30 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 30 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(7/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=- \frac {2 A a^{4}}{5 x^{\frac {5}{2}}} - \frac {8 A a^{3} b}{3 x^{\frac {3}{2}}} - \frac {12 A a^{2} b^{2}}{\sqrt {x}} + 8 A a b^{3} \sqrt {x} + \frac {2 A b^{4} x^{\frac {3}{2}}}{3} - \frac {2 B a^{4}}{3 x^{\frac {3}{2}}} - \frac {8 B a^{3} b}{\sqrt {x}} + 12 B a^{2} b^{2} \sqrt {x} + \frac {8 B a b^{3} x^{\frac {3}{2}}}{3} + \frac {2 B b^{4} x^{\frac {5}{2}}}{5} \]

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**(7/2),x)

[Out]

-2*A*a**4/(5*x**(5/2)) - 8*A*a**3*b/(3*x**(3/2)) - 12*A*a**2*b**2/sqrt(x) + 8*A*a*b**3*sqrt(x) + 2*A*b**4*x**(
3/2)/3 - 2*B*a**4/(3*x**(3/2)) - 8*B*a**3*b/sqrt(x) + 12*B*a**2*b**2*sqrt(x) + 8*B*a*b**3*x**(3/2)/3 + 2*B*b**
4*x**(5/2)/5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{5} \, B b^{4} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {3}{2}} + 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{4} + 30 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(7/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{5} \, B b^{4} x^{\frac {5}{2}} + \frac {8}{3} \, B a b^{3} x^{\frac {3}{2}} + \frac {2}{3} \, A b^{4} x^{\frac {3}{2}} + 12 \, B a^{2} b^{2} \sqrt {x} + 8 \, A a b^{3} \sqrt {x} - \frac {2 \, {\left (60 \, B a^{3} b x^{2} + 90 \, A a^{2} b^{2} x^{2} + 5 \, B a^{4} x + 20 \, A a^{3} b x + 3 \, A a^{4}\right )}}{15 \, x^{\frac {5}{2}}} \]

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(7/2),x, algorithm="giac")

[Out]

2/5*B*b^4*x^(5/2) + 8/3*B*a*b^3*x^(3/2) + 2/3*A*b^4*x^(3/2) + 12*B*a^2*b^2*sqrt(x) + 8*A*a*b^3*sqrt(x) - 2/15*
(60*B*a^3*b*x^2 + 90*A*a^2*b^2*x^2 + 5*B*a^4*x + 20*A*a^3*b*x + 3*A*a^4)/x^(5/2)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^{7/2}} \, dx=x^{3/2}\,\left (\frac {2\,A\,b^4}{3}+\frac {8\,B\,a\,b^3}{3}\right )-\frac {x\,\left (\frac {2\,B\,a^4}{3}+\frac {8\,A\,b\,a^3}{3}\right )+\frac {2\,A\,a^4}{5}+x^2\,\left (8\,B\,a^3\,b+12\,A\,a^2\,b^2\right )}{x^{5/2}}+\frac {2\,B\,b^4\,x^{5/2}}{5}+4\,a\,b^2\,\sqrt {x}\,\left (2\,A\,b+3\,B\,a\right ) \]

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/x^(7/2),x)

[Out]

x^(3/2)*((2*A*b^4)/3 + (8*B*a*b^3)/3) - (x*((2*B*a^4)/3 + (8*A*a^3*b)/3) + (2*A*a^4)/5 + x^2*(12*A*a^2*b^2 + 8
*B*a^3*b))/x^(5/2) + (2*B*b^4*x^(5/2))/5 + 4*a*b^2*x^(1/2)*(2*A*b + 3*B*a)