\(\int \frac {(a+b x^2) (A+B x^2)}{\sqrt {x}} \, dx\) [347]

Optimal result

Integrand size = 20, antiderivative size = 37 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 a A \sqrt {x}+\frac {2}{5} (A b+a B) x^{5/2}+\frac {2}{9} b B x^{9/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{5} x^{5/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{9} b B x^{9/2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{45} \sqrt {x} \left (45 a A+9 A b x^2+9 a B x^2+5 b B x^4\right ) \]

[In]

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

[Out]

(2*Sqrt[x]*(45*a*A + 9*A*b*x^2 + 9*a*B*x^2 + 5*b*B*x^4))/45

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {2 \left (A b +B a \right ) x^{\frac {5}{2}}}{5}+\frac {2 b B \,x^{\frac {9}{2}}}{9}+2 a A \sqrt {x}\)	\(28\)
default	\(\frac {2 \left (A b +B a \right ) x^{\frac {5}{2}}}{5}+\frac {2 b B \,x^{\frac {9}{2}}}{9}+2 a A \sqrt {x}\)	\(28\)
trager	\(\left (\frac {2}{9} b B \,x^{4}+\frac {2}{5} A b \,x^{2}+\frac {2}{5} B a \,x^{2}+2 A a \right ) \sqrt {x}\)	\(31\)
gosper	\(\frac {2 \sqrt {x}\, \left (5 b B \,x^{4}+9 A b \,x^{2}+9 B a \,x^{2}+45 A a \right )}{45}\)	\(32\)
risch	\(\frac {2 \sqrt {x}\, \left (5 b B \,x^{4}+9 A b \,x^{2}+9 B a \,x^{2}+45 A a \right )}{45}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{45} \, {\left (5 \, B b x^{4} + 9 \, {\left (B a + A b\right )} x^{2} + 45 \, A a\right )} \sqrt {x} \]

[In]

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

[Out]

2/45*(5*B*b*x^4 + 9*(B*a + A*b)*x^2 + 45*A*a)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 A a \sqrt {x} + \frac {2 A b x^{\frac {5}{2}}}{5} + \frac {2 B a x^{\frac {5}{2}}}{5} + \frac {2 B b x^{\frac {9}{2}}}{9} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{5} \, {\left (B a + A b\right )} x^{\frac {5}{2}} + 2 \, A a \sqrt {x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{5} \, B a x^{\frac {5}{2}} + \frac {2}{5} \, A b x^{\frac {5}{2}} + 2 \, A a \sqrt {x} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2\,\sqrt {x}\,\left (45\,A\,a+9\,A\,b\,x^2+9\,B\,a\,x^2+5\,B\,b\,x^4\right )}{45} \]

[In]

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

[Out]

(2*x^(1/2)*(45*A*a + 9*A*b*x^2 + 9*B*a*x^2 + 5*B*b*x^4))/45