\(\int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx\) [324]

Optimal result

Integrand size = 16, antiderivative size = 35 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=-\frac {2 a A}{\sqrt {x}}+2 (A b+a B) \sqrt {x}+\frac {2}{3} b B x^{3/2} \]

[Out]

2/3*b*B*x^(3/2)-2*a*A/x^(1/2)+2*(A*b+B*a)*x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=2 \sqrt {x} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{3} b B x^{3/2} \]

[In]

Int[((a + b*x)*(A + B*x))/x^(3/2),x]

[Out]

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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a A}{x^{3/2}}+\frac {A b+a B}{\sqrt {x}}+b B \sqrt {x}\right ) \, dx \\ & = -\frac {2 a A}{\sqrt {x}}+2 (A b+a B) \sqrt {x}+\frac {2}{3} b B x^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=-\frac {2 \left (3 a A-3 A b x-3 a B x-b B x^2\right )}{3 \sqrt {x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80

method	result	size
gosper	\(-\frac {2 \left (-b B \,x^{2}-3 A b x -3 B a x +3 A a \right )}{3 \sqrt {x}}\)	\(28\)
trager	\(-\frac {2 \left (-b B \,x^{2}-3 A b x -3 B a x +3 A a \right )}{3 \sqrt {x}}\)	\(28\)
risch	\(-\frac {2 \left (-b B \,x^{2}-3 A b x -3 B a x +3 A a \right )}{3 \sqrt {x}}\)	\(28\)
derivativedivides	\(\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}+2 B a \sqrt {x}-\frac {2 a A}{\sqrt {x}}\)	\(30\)
default	\(\frac {2 b B \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}+2 B a \sqrt {x}-\frac {2 a A}{\sqrt {x}}\)	\(30\)

[In]

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

[Out]

-2/3*(-B*b*x^2-3*A*b*x-3*B*a*x+3*A*a)/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=\frac {2 \, {\left (B b x^{2} - 3 \, A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, \sqrt {x}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=- \frac {2 A a}{\sqrt {x}} + 2 A b \sqrt {x} + 2 B a \sqrt {x} + \frac {2 B b x^{\frac {3}{2}}}{3} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=\frac {2}{3} \, B b x^{\frac {3}{2}} - \frac {2 \, A a}{\sqrt {x}} + 2 \, {\left (B a + A b\right )} \sqrt {x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=\frac {2}{3} \, B b x^{\frac {3}{2}} + 2 \, B a \sqrt {x} + 2 \, A b \sqrt {x} - \frac {2 \, A a}{\sqrt {x}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x) (A+B x)}{x^{3/2}} \, dx=\frac {6\,A\,b\,x-6\,A\,a+6\,B\,a\,x+2\,B\,b\,x^2}{3\,\sqrt {x}} \]

[In]

int(((A + B*x)*(a + b*x))/x^(3/2),x)

[Out]

(6*A*b*x - 6*A*a + 6*B*a*x + 2*B*b*x^2)/(3*x^(1/2))