3.2249 \(\int \frac{A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=89 \[ \frac{2 \sqrt{d+e x} (a B e-2 A b e+b B d)}{e \sqrt{a+b x} (b d-a e)^2}-\frac{2 (B d-A e)}{e \sqrt{a+b x} \sqrt{d+e x} (b d-a e)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0539108, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {78, 37} \[ \frac{2 \sqrt{d+e x} (a B e-2 A b e+b B d)}{e \sqrt{a+b x} (b d-a e)^2}-\frac{2 (B d-A e)}{e \sqrt{a+b x} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e))/(e*(b*d - a*e)*Sqrt[a + b*x]*Sqrt[d + e*x]) + (2*(b*B*d - 2*A*b*e + a*B*e)*Sqrt[d + e*x])/(e*
(b*d - a*e)^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{A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx &=-\frac{2 (B d-A e)}{e (b d-a e) \sqrt{a+b x} \sqrt{d+e x}}-\frac{(b B d-2 A b e+a B e) \int \frac{1}{(a+b x)^{3/2} \sqrt{d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac{2 (B d-A e)}{e (b d-a e) \sqrt{a+b x} \sqrt{d+e x}}+\frac{2 (b B d-2 A b e+a B e) \sqrt{d+e x}}{e (b d-a e)^2 \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0340073, size = 61, normalized size = 0.69 \[ \frac{2 B (2 a d+a e x+b d x)-2 A (a e+b (d+2 e x))}{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 72, normalized size = 0.8 \begin{align*} -2\,{\frac{2\,Abex-Baex-Bbdx+Aae+Abd-2\,Bad}{\sqrt{bx+a}\sqrt{ex+d} \left ({a}^{2}{e}^{2}-2\,bead+{b}^{2}{d}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*A*b*e*x-B*a*e*x-B*b*d*x+A*a*e+A*b*d-2*B*a*d)/(b*x+a)^(1/2)/(e*x+d)^(1/2)/(a^2*e^2-2*a*b*d*e+b^2*d^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((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.31788, size = 305, normalized size = 3.43 \begin{align*} -\frac{2 \,{\left (A a e -{\left (2 \, B a - A b\right )} d -{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)/((a + b*x)**(3/2)*(d + e*x)**(3/2)), x)

________________________________________________________________________________________

Giac [B]  time = 2.4039, size = 227, normalized size = 2.55 \begin{align*} \frac{2 \,{\left (B b^{2} d - A b^{2} e\right )} \sqrt{b x + a}}{{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} + \frac{4 \,{\left (B a b^{\frac{3}{2}} e^{\frac{1}{2}} - A b^{\frac{5}{2}} e^{\frac{1}{2}}\right )}}{{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{\left (b d{\left | b \right |} - a{\left | b \right |} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(B*b^2*d - A*b^2*e)*sqrt(b*x + a)/((b^2*d^2*abs(b) - 2*a*b*d*abs(b)*e + a^2*abs(b)*e^2)*sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)) + 4*(B*a*b^(3/2)*e^(1/2) - A*b^(5/2)*e^(1/2))/((b^2*d - a*b*e - (sqrt(b*x + a)*sqrt(b)*e^(1/2
) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)*(b*d*abs(b) - a*abs(b)*e))