\(\int \frac {(a+b x^2)^2 (c+d x^2)}{x^{7/2}} \, dx\) [398]

Optimal result

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

[Out]

-2/5*a^2*c/x^(5/2)+2/3*b*(2*a*d+b*c)*x^(3/2)+2/7*b^2*d*x^(7/2)-2*a*(a*d+2*b*c)/x^(1/2)

Rubi [A] (verified)

Time = 0.04 (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} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx=-\frac {2 a^2 c}{5 x^{5/2}}+\frac {2}{3} b x^{3/2} (2 a d+b c)-\frac {2 a (a d+2 b c)}{\sqrt {x}}+\frac {2}{7} b^2 d x^{7/2} \]

[In]

Int[((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx=-\frac {2 \left (21 a^2 c+210 a b c x^2+105 a^2 d x^2-35 b^2 c x^4-70 a b d x^4-15 b^2 d x^6\right )}{105 x^{5/2}} \]

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x]

[Out]

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

Maple [A] (verified)

Time = 2.71 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx=- \frac {2 a^{2} c}{5 x^{\frac {5}{2}}} - \frac {2 a^{2} d}{\sqrt {x}} - \frac {4 a b c}{\sqrt {x}} + \frac {4 a b d x^{\frac {3}{2}}}{3} + \frac {2 b^{2} c x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d x^{\frac {7}{2}}}{7} \]

[In]

integrate((b*x**2+a)**2*(d*x**2+c)/x**(7/2),x)

[Out]

-2*a**2*c/(5*x**(5/2)) - 2*a**2*d/sqrt(x) - 4*a*b*c/sqrt(x) + 4*a*b*d*x**(3/2)/3 + 2*b**2*c*x**(3/2)/3 + 2*b**
2*d*x**(7/2)/7

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx=\frac {2}{7} \, b^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (a^{2} c + 5 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]

[In]

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

[Out]

2/7*b^2*d*x^(7/2) + 2/3*(b^2*c + 2*a*b*d)*x^(3/2) - 2/5*(a^2*c + 5*(2*a*b*c + a^2*d)*x^2)/x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx=\frac {2}{7} \, b^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} c x^{\frac {3}{2}} + \frac {4}{3} \, a b d x^{\frac {3}{2}} - \frac {2 \, {\left (10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, x^{\frac {5}{2}}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx=-\frac {210\,d\,a^2\,x^2+42\,c\,a^2-140\,d\,a\,b\,x^4+420\,c\,a\,b\,x^2-30\,d\,b^2\,x^6-70\,c\,b^2\,x^4}{105\,x^{5/2}} \]

[In]

int(((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x)

[Out]

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