\(\int \frac {(a+b x^2)^2 (A+B x^2)}{x^{7/2}} \, dx\) [358]

Optimal result

Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=-\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (2 A b+a B)}{\sqrt {x}}+\frac {2}{3} b (A b+2 a B) x^{3/2}+\frac {2}{7} b^2 B x^{7/2} \]

[Out]

-2/5*a^2*A/x^(5/2)+2/3*b*(A*b+2*B*a)*x^(3/2)+2/7*b^2*B*x^(7/2)-2*a*(2*A*b+B*a)/x^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=-\frac {2 a^2 A}{5 x^{5/2}}+\frac {2}{3} b x^{3/2} (2 a B+A b)-\frac {2 a (a B+2 A b)}{\sqrt {x}}+\frac {2}{7} b^2 B x^{7/2} \]

[In]

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

[Out]

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=-\frac {2 \left (21 a^2 A+210 a A b x^2+105 a^2 B x^2-35 A b^2 x^4-70 a b B x^4-15 b^2 B x^6\right )}{105 x^{5/2}} \]

[In]

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

[Out]

(-2*(21*a^2*A + 210*a*A*b*x^2 + 105*a^2*B*x^2 - 35*A*b^2*x^4 - 70*a*b*B*x^4 - 15*b^2*B*x^6))/(105*x^(5/2))

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {2 b^{2} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{2} x^{\frac {3}{2}}}{3}+\frac {4 B a b \,x^{\frac {3}{2}}}{3}-\frac {2 a^{2} A}{5 x^{\frac {5}{2}}}-\frac {2 a \left (2 A b +B a \right )}{\sqrt {x}}\)	\(51\)
default	\(\frac {2 b^{2} B \,x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{2} x^{\frac {3}{2}}}{3}+\frac {4 B a b \,x^{\frac {3}{2}}}{3}-\frac {2 a^{2} A}{5 x^{\frac {5}{2}}}-\frac {2 a \left (2 A b +B a \right )}{\sqrt {x}}\)	\(51\)
gosper	\(-\frac {2 \left (-15 b^{2} B \,x^{6}-35 A \,b^{2} x^{4}-70 B a b \,x^{4}+210 a A b \,x^{2}+105 a^{2} B \,x^{2}+21 a^{2} A \right )}{105 x^{\frac {5}{2}}}\)	\(56\)
trager	\(-\frac {2 \left (-15 b^{2} B \,x^{6}-35 A \,b^{2} x^{4}-70 B a b \,x^{4}+210 a A b \,x^{2}+105 a^{2} B \,x^{2}+21 a^{2} A \right )}{105 x^{\frac {5}{2}}}\)	\(56\)
risch	\(-\frac {2 \left (-15 b^{2} B \,x^{6}-35 A \,b^{2} x^{4}-70 B a b \,x^{4}+210 a A b \,x^{2}+105 a^{2} B \,x^{2}+21 a^{2} A \right )}{105 x^{\frac {5}{2}}}\)	\(56\)

[In]

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

[Out]

2/7*b^2*B*x^(7/2)+2/3*A*b^2*x^(3/2)+4/3*B*a*b*x^(3/2)-2/5*a^2*A/x^(5/2)-2*a*(2*A*b+B*a)/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=\frac {2 \, {\left (15 \, B b^{2} x^{6} + 35 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 21 \, A a^{2} - 105 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{105 \, x^{\frac {5}{2}}} \]

[In]

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

[Out]

2/105*(15*B*b^2*x^6 + 35*(2*B*a*b + A*b^2)*x^4 - 21*A*a^2 - 105*(B*a^2 + 2*A*a*b)*x^2)/x^(5/2)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=- \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a b}{\sqrt {x}} + \frac {2 A b^{2} x^{\frac {3}{2}}}{3} - \frac {2 B a^{2}}{\sqrt {x}} + \frac {4 B a b x^{\frac {3}{2}}}{3} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=\frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {2}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (A a^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=\frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{3} \, B a b x^{\frac {3}{2}} + \frac {2}{3} \, A b^{2} x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + A a^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.79 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx=-\frac {210\,B\,a^2\,x^2+42\,A\,a^2-140\,B\,a\,b\,x^4+420\,A\,a\,b\,x^2-30\,B\,b^2\,x^6-70\,A\,b^2\,x^4}{105\,x^{5/2}} \]

[In]

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

[Out]

-(42*A*a^2 + 210*B*a^2*x^2 - 70*A*b^2*x^4 - 30*B*b^2*x^6 + 420*A*a*b*x^2 - 140*B*a*b*x^4)/(105*x^(5/2))