\(\int \frac {(a+b x^2)^2 (c+d x^2)}{x^{5/2}} \, dx\) [397]

Optimal result

Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^2 c}{3 x^{3/2}}+2 a (2 b c+a d) \sqrt {x}+\frac {2}{5} b (b c+2 a d) x^{5/2}+\frac {2}{9} b^2 d x^{9/2} \]

[Out]

-2/3*a^2*c/x^(3/2)+2/5*b*(2*a*d+b*c)*x^(5/2)+2/9*b^2*d*x^(9/2)+2*a*(a*d+2*b*c)*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} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^2 c}{3 x^{3/2}}+\frac {2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt {x} (a d+2 b c)+\frac {2}{9} b^2 d x^{9/2} \]

[In]

Int[((a + b*x^2)^2*(c + d*x^2))/x^(5/2),x]

[Out]

(-2*a^2*c)/(3*x^(3/2)) + 2*a*(2*b*c + a*d)*Sqrt[x] + (2*b*(b*c + 2*a*d)*x^(5/2))/5 + (2*b^2*d*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^2 c}{x^{5/2}}+\frac {a (2 b c+a d)}{\sqrt {x}}+b (b c+2 a d) x^{3/2}+b^2 d x^{7/2}\right ) \, dx \\ & = -\frac {2 a^2 c}{3 x^{3/2}}+2 a (2 b c+a d) \sqrt {x}+\frac {2}{5} b (b c+2 a d) x^{5/2}+\frac {2}{9} b^2 d x^{9/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=-\frac {2 \left (15 a^2 c-90 a b c x^2-45 a^2 d x^2-9 b^2 c x^4-18 a b d x^4-5 b^2 d x^6\right )}{45 x^{3/2}} \]

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(5/2),x]

[Out]

(-2*(15*a^2*c - 90*a*b*c*x^2 - 45*a^2*d*x^2 - 9*b^2*c*x^4 - 18*a*b*d*x^4 - 5*b^2*d*x^6))/(45*x^(3/2))

Maple [A] (verified)

Time = 2.68 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {2 b^{2} d \,x^{\frac {9}{2}}}{9}+\frac {4 a b d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c \,x^{\frac {5}{2}}}{5}+2 a^{2} d \sqrt {x}+4 a b c \sqrt {x}-\frac {2 a^{2} c}{3 x^{\frac {3}{2}}}\)	\(54\)
default	\(\frac {2 b^{2} d \,x^{\frac {9}{2}}}{9}+\frac {4 a b d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c \,x^{\frac {5}{2}}}{5}+2 a^{2} d \sqrt {x}+4 a b c \sqrt {x}-\frac {2 a^{2} c}{3 x^{\frac {3}{2}}}\)	\(54\)
gosper	\(-\frac {2 \left (-5 b^{2} d \,x^{6}-18 a b d \,x^{4}-9 b^{2} c \,x^{4}-45 a^{2} d \,x^{2}-90 a b c \,x^{2}+15 a^{2} c \right )}{45 x^{\frac {3}{2}}}\)	\(56\)
trager	\(-\frac {2 \left (-5 b^{2} d \,x^{6}-18 a b d \,x^{4}-9 b^{2} c \,x^{4}-45 a^{2} d \,x^{2}-90 a b c \,x^{2}+15 a^{2} c \right )}{45 x^{\frac {3}{2}}}\)	\(56\)
risch	\(-\frac {2 \left (-5 b^{2} d \,x^{6}-18 a b d \,x^{4}-9 b^{2} c \,x^{4}-45 a^{2} d \,x^{2}-90 a b c \,x^{2}+15 a^{2} c \right )}{45 x^{\frac {3}{2}}}\)	\(56\)

[In]

int((b*x^2+a)^2*(d*x^2+c)/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/9*b^2*d*x^(9/2)+4/5*a*b*d*x^(5/2)+2/5*b^2*c*x^(5/2)+2*a^2*d*x^(1/2)+4*a*b*c*x^(1/2)-2/3*a^2*c/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=\frac {2 \, {\left (5 \, b^{2} d x^{6} + 9 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 45 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{45 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)/x^(5/2),x, algorithm="fricas")

[Out]

2/45*(5*b^2*d*x^6 + 9*(b^2*c + 2*a*b*d)*x^4 - 15*a^2*c + 45*(2*a*b*c + a^2*d)*x^2)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=- \frac {2 a^{2} c}{3 x^{\frac {3}{2}}} + 2 a^{2} d \sqrt {x} + 4 a b c \sqrt {x} + \frac {4 a b d x^{\frac {5}{2}}}{5} + \frac {2 b^{2} c x^{\frac {5}{2}}}{5} + \frac {2 b^{2} d x^{\frac {9}{2}}}{9} \]

[In]

integrate((b*x**2+a)**2*(d*x**2+c)/x**(5/2),x)

[Out]

-2*a**2*c/(3*x**(3/2)) + 2*a**2*d*sqrt(x) + 4*a*b*c*sqrt(x) + 4*a*b*d*x**(5/2)/5 + 2*b**2*c*x**(5/2)/5 + 2*b**
2*d*x**(9/2)/9

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=\frac {2}{9} \, b^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {5}{2}} - \frac {2 \, a^{2} c}{3 \, x^{\frac {3}{2}}} + 2 \, {\left (2 \, a b c + a^{2} d\right )} \sqrt {x} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)/x^(5/2),x, algorithm="maxima")

[Out]

2/9*b^2*d*x^(9/2) + 2/5*(b^2*c + 2*a*b*d)*x^(5/2) - 2/3*a^2*c/x^(3/2) + 2*(2*a*b*c + a^2*d)*sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=\frac {2}{9} \, b^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c x^{\frac {5}{2}} + \frac {4}{5} \, a b d x^{\frac {5}{2}} + 4 \, a b c \sqrt {x} + 2 \, a^{2} d \sqrt {x} - \frac {2 \, a^{2} c}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)/x^(5/2),x, algorithm="giac")

[Out]

2/9*b^2*d*x^(9/2) + 2/5*b^2*c*x^(5/2) + 4/5*a*b*d*x^(5/2) + 4*a*b*c*sqrt(x) + 2*a^2*d*sqrt(x) - 2/3*a^2*c/x^(3
/2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx=\sqrt {x}\,\left (2\,d\,a^2+4\,b\,c\,a\right )+x^{5/2}\,\left (\frac {2\,c\,b^2}{5}+\frac {4\,a\,d\,b}{5}\right )-\frac {2\,a^2\,c}{3\,x^{3/2}}+\frac {2\,b^2\,d\,x^{9/2}}{9} \]

[In]

int(((a + b*x^2)^2*(c + d*x^2))/x^(5/2),x)

[Out]

x^(1/2)*(2*a^2*d + 4*a*b*c) + x^(5/2)*((2*b^2*c)/5 + (4*a*b*d)/5) - (2*a^2*c)/(3*x^(3/2)) + (2*b^2*d*x^(9/2))/
9