\(\int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) [779]

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {2 c}{d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 (b c+a d) \sqrt {c+d x}}{d (b c-a d)^2 \sqrt {a+b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {c+d x} (a d+b c)}{d \sqrt {a+b x} (b c-a d)^2}-\frac {2 c}{d \sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \]

[In]

Int[x/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(-2*c)/(d*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + (2*(b*c + a*d)*Sqrt[c + d*x])/(d*(b*c - a*d)^2*Sqrt[a + b
*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {(b c+a d) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{d (b c-a d)} \\ & = -\frac {2 c}{d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 (b c+a d) \sqrt {c+d x}}{d (b c-a d)^2 \sqrt {a+b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.57 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 (2 a c+b c x+a d x)}{(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}} \]

[In]

Integrate[x/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(2*(2*a*c + b*c*x + a*d*x))/((b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {2 a d x +2 b c x +4 a c}{\left (a d -b c \right )^{2} \sqrt {b x +a}\, \sqrt {d x +c}}\)	\(40\)
gosper	\(\frac {2 a d x +2 b c x +4 a c}{\sqrt {b x +a}\, \sqrt {d x +c}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(53\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.69 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(x/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Integral(x/((a + b*x)**(3/2)*(c + d*x)**(3/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (67) = 134\).

Time = 0.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.96 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {\sqrt {b x + a} b^{3} c}{{\left (b^{2} c^{2} {\left | b \right |} - 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {2 \, \sqrt {b d} a b^{2}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} {\left (b c {\left | b \right |} - a d {\left | b \right |}\right )}}\right )}}{b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {\left (\frac {x\,\left (2\,a\,d+2\,b\,c\right )}{d\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,a\,c}{d\,{\left (a\,d-b\,c\right )}^2}\right )\,\sqrt {c+d\,x}}{x\,\sqrt {a+b\,x}+\frac {c\,\sqrt {a+b\,x}}{d}} \]

[In]

int(x/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x)

[Out]

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