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

Optimal. Leaf size=75 \[ \frac{2 \sqrt{c+d x} (a d+b c)}{d \sqrt{a+b x} (b c-a d)^2}-\frac{2 c}{d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.0220603, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{2 \sqrt{c+d x} (a d+b c)}{d \sqrt{a+b x} (b c-a d)^2}-\frac{2 c}{d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[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 78

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] || LtQ
[p, n]))))

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]

Rubi steps

\begin{align*} \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{(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]  time = 0.0150238, size = 43, normalized size = 0.57 \[ \frac{2 (2 a c+a d x+b c x)}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[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]  time = 0.004, size = 53, normalized size = 0.7 \begin{align*} 2\,{\frac{adx+bcx+2\,ac}{\sqrt{bx+a}\sqrt{dx+c} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.8559, size = 261, normalized size = 3.48 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [B]  time = 1.42003, size = 198, normalized size = 2.64 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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