\(\int \frac {(a+b x^2)^2 (A+B x^2)}{\sqrt {x}} \, dx\) [355]

Optimal result

Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 a^2 A \sqrt {x}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{9} b (A b+2 a B) x^{9/2}+\frac {2}{13} b^2 B x^{13/2} \]

[Out]

2/5*a*(2*A*b+B*a)*x^(5/2)+2/9*b*(A*b+2*B*a)*x^(9/2)+2/13*b^2*B*x^(13/2)+2*a^2*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} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 a^2 A \sqrt {x}+\frac {2}{9} b x^{9/2} (2 a B+A b)+\frac {2}{5} a x^{5/2} (a B+2 A b)+\frac {2}{13} b^2 B x^{13/2} \]

[In]

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

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(9/2))/9 + (2*b^2*B*x^(13/2))/13

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}{\sqrt {x}}+a (2 A b+a B) x^{3/2}+b (A b+2 a B) x^{7/2}+b^2 B x^{11/2}\right ) \, dx \\ & = 2 a^2 A \sqrt {x}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{9} b (A b+2 a B) x^{9/2}+\frac {2}{13} b^2 B x^{13/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{585} \sqrt {x} \left (117 a^2 \left (5 A+B x^2\right )+26 a b x^2 \left (9 A+5 B x^2\right )+5 b^2 x^4 \left (13 A+9 B x^2\right )\right ) \]

[In]

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

[Out]

(2*Sqrt[x]*(117*a^2*(5*A + B*x^2) + 26*a*b*x^2*(9*A + 5*B*x^2) + 5*b^2*x^4*(13*A + 9*B*x^2)))/585

Maple [A] (verified)

Time = 2.61 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {2 b^{2} B \,x^{\frac {13}{2}}}{13}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {5}{2}}}{5}+2 a^{2} A \sqrt {x}\)	\(52\)
default	\(\frac {2 b^{2} B \,x^{\frac {13}{2}}}{13}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {5}{2}}}{5}+2 a^{2} A \sqrt {x}\)	\(52\)
trager	\(\left (\frac {2}{13} b^{2} B \,x^{6}+\frac {2}{9} A \,b^{2} x^{4}+\frac {4}{9} B a b \,x^{4}+\frac {4}{5} a A b \,x^{2}+\frac {2}{5} a^{2} B \,x^{2}+2 a^{2} A \right ) \sqrt {x}\)	\(55\)
gosper	\(\frac {2 \sqrt {x}\, \left (45 b^{2} B \,x^{6}+65 A \,b^{2} x^{4}+130 B a b \,x^{4}+234 a A b \,x^{2}+117 a^{2} B \,x^{2}+585 a^{2} A \right )}{585}\)	\(56\)
risch	\(\frac {2 \sqrt {x}\, \left (45 b^{2} B \,x^{6}+65 A \,b^{2} x^{4}+130 B a b \,x^{4}+234 a A b \,x^{2}+117 a^{2} B \,x^{2}+585 a^{2} A \right )}{585}\)	\(56\)

[In]

int((b*x^2+a)^2*(B*x^2+A)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/13*b^2*B*x^(13/2)+2/9*(A*b^2+2*B*a*b)*x^(9/2)+2/5*(2*A*a*b+B*a^2)*x^(5/2)+2*a^2*A*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{585} \, {\left (45 \, B b^{2} x^{6} + 65 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 585 \, A a^{2} + 117 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt {x} \]

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(1/2),x, algorithm="fricas")

[Out]

2/585*(45*B*b^2*x^6 + 65*(2*B*a*b + A*b^2)*x^4 + 585*A*a^2 + 117*(B*a^2 + 2*A*a*b)*x^2)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 A a^{2} \sqrt {x} + \frac {4 A a b x^{\frac {5}{2}}}{5} + \frac {2 A b^{2} x^{\frac {9}{2}}}{9} + \frac {2 B a^{2} x^{\frac {5}{2}}}{5} + \frac {4 B a b x^{\frac {9}{2}}}{9} + \frac {2 B b^{2} x^{\frac {13}{2}}}{13} \]

[In]

integrate((b*x**2+a)**2*(B*x**2+A)/x**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(5/2)/5 + 2*A*b**2*x**(9/2)/9 + 2*B*a**2*x**(5/2)/5 + 4*B*a*b*x**(9/2)/9 + 2*B*b
**2*x**(13/2)/13

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {9}{2}} + 2 \, A a^{2} \sqrt {x} + \frac {2}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {5}{2}} \]

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(1/2),x, algorithm="maxima")

[Out]

2/13*B*b^2*x^(13/2) + 2/9*(2*B*a*b + A*b^2)*x^(9/2) + 2*A*a^2*sqrt(x) + 2/5*(B*a^2 + 2*A*a*b)*x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {4}{9} \, B a b x^{\frac {9}{2}} + \frac {2}{9} \, A b^{2} x^{\frac {9}{2}} + \frac {2}{5} \, B a^{2} x^{\frac {5}{2}} + \frac {4}{5} \, A a b x^{\frac {5}{2}} + 2 \, A a^{2} \sqrt {x} \]

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(1/2),x, algorithm="giac")

[Out]

2/13*B*b^2*x^(13/2) + 4/9*B*a*b*x^(9/2) + 2/9*A*b^2*x^(9/2) + 2/5*B*a^2*x^(5/2) + 4/5*A*a*b*x^(5/2) + 2*A*a^2*
sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt {x}} \, dx=x^{5/2}\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )+x^{9/2}\,\left (\frac {2\,A\,b^2}{9}+\frac {4\,B\,a\,b}{9}\right )+2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,b^2\,x^{13/2}}{13} \]

[In]

int(((A + B*x^2)*(a + b*x^2)^2)/x^(1/2),x)

[Out]

x^(5/2)*((2*B*a^2)/5 + (4*A*a*b)/5) + x^(9/2)*((2*A*b^2)/9 + (4*B*a*b)/9) + 2*A*a^2*x^(1/2) + (2*B*b^2*x^(13/2
))/13