\(\int \frac {(a+b x^2) (A+B x^2)}{x^{3/2}} \, dx\) [348]

Optimal result

Integrand size = 20, antiderivative size = 37 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=-\frac {2 a A}{\sqrt {x}}+\frac {2}{3} (A b+a B) x^{3/2}+\frac {2}{7} b B x^{7/2} \]

[Out]

2/3*(A*b+B*a)*x^(3/2)+2/7*b*B*x^(7/2)-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} \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2}{3} x^{3/2} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{7} b B x^{7/2} \]

[In]

Int[((a + b*x^2)*(A + B*x^2))/x^(3/2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2 \left (-21 a A+7 A b x^2+7 a B x^2+3 b B x^4\right )}{21 \sqrt {x}} \]

[In]

Integrate[((a + b*x^2)*(A + B*x^2))/x^(3/2),x]

[Out]

(2*(-21*a*A + 7*A*b*x^2 + 7*a*B*x^2 + 3*b*B*x^4))/(21*Sqrt[x])

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {2 b B \,x^{\frac {7}{2}}}{7}+\frac {2 A b \,x^{\frac {3}{2}}}{3}+\frac {2 B a \,x^{\frac {3}{2}}}{3}-\frac {2 a A}{\sqrt {x}}\)	\(30\)
default	\(\frac {2 b B \,x^{\frac {7}{2}}}{7}+\frac {2 A b \,x^{\frac {3}{2}}}{3}+\frac {2 B a \,x^{\frac {3}{2}}}{3}-\frac {2 a A}{\sqrt {x}}\)	\(30\)
gosper	\(-\frac {2 \left (-3 b B \,x^{4}-7 A b \,x^{2}-7 B a \,x^{2}+21 A a \right )}{21 \sqrt {x}}\)	\(32\)
trager	\(-\frac {2 \left (-3 b B \,x^{4}-7 A b \,x^{2}-7 B a \,x^{2}+21 A a \right )}{21 \sqrt {x}}\)	\(32\)
risch	\(-\frac {2 \left (-3 b B \,x^{4}-7 A b \,x^{2}-7 B a \,x^{2}+21 A a \right )}{21 \sqrt {x}}\)	\(32\)

[In]

int((b*x^2+a)*(B*x^2+A)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/7*b*B*x^(7/2)+2/3*A*b*x^(3/2)+2/3*B*a*x^(3/2)-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 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2 \, {\left (3 \, B b x^{4} + 7 \, {\left (B a + A b\right )} x^{2} - 21 \, A a\right )}}{21 \, \sqrt {x}} \]

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*B*b*x^4 + 7*(B*a + A*b)*x^2 - 21*A*a)/sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=- \frac {2 A a}{\sqrt {x}} + \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 B a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {7}{2}}}{7} \]

[In]

integrate((b*x**2+a)*(B*x**2+A)/x**(3/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{3} \, {\left (B a + A b\right )} x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} \]

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} \]

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.86 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {14\,A\,b\,x^2-42\,A\,a+14\,B\,a\,x^2+6\,B\,b\,x^4}{21\,\sqrt {x}} \]

[In]

int(((A + B*x^2)*(a + b*x^2))/x^(3/2),x)

[Out]

(14*A*b*x^2 - 42*A*a + 14*B*a*x^2 + 6*B*b*x^4)/(21*x^(1/2))