3.1747 \(\int \frac{A+B x}{(a+b x) (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2 (B d-A e)}{e \sqrt{d+e x} (b d-a e)}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0562895, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {78, 63, 208} \[ -\frac{2 (B d-A e)}{e \sqrt{d+e x} (b d-a e)}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{A+B x}{(a+b x) (d+e x)^{3/2}} \, dx &=-\frac{2 (B d-A e)}{e (b d-a e) \sqrt{d+e x}}+\frac{(A b-a B) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{b d-a e}\\ &=-\frac{2 (B d-A e)}{e (b d-a e) \sqrt{d+e x}}+\frac{(2 (A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e (b d-a e)}\\ &=-\frac{2 (B d-A e)}{e (b d-a e) \sqrt{d+e x}}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.163973, size = 97, normalized size = 1.1 \[ \frac{2 \left (\frac{(a e-b d) (B d-A e)}{\sqrt{d+e x}}+\frac{e (a B-A b) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b}}\right )}{e (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 142, normalized size = 1.6 \begin{align*} -2\,{\frac{Ab}{ \left ( ae-bd \right ) \sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{Ba}{ \left ( ae-bd \right ) \sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{A}{ \left ( ae-bd \right ) \sqrt{ex+d}}}+2\,{\frac{Bd}{e \left ( ae-bd \right ) \sqrt{ex+d}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*e-b*d)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*b+2/(a*e-b*d)/((a*e-b*d)*b)^(1/
2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a-2/(a*e-b*d)/(e*x+d)^(1/2)*A+2/e/(a*e-b*d)/(e*x+d)^(1/2)*B*d

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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)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.46322, size = 767, normalized size = 8.72 \begin{align*} \left [\frac{{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (B b^{2} d^{2} + A a b e^{2} -{\left (B a b + A b^{2}\right )} d e\right )} \sqrt{e x + d}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} +{\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}, -\frac{2 \,{\left ({\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) +{\left (B b^{2} d^{2} + A a b e^{2} -{\left (B a b + A b^{2}\right )} d e\right )} \sqrt{e x + d}\right )}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} +{\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(((B*a - A*b)*e^2*x + (B*a - A*b)*d*e)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e + 2*sqrt(b^2*d - a*b*e)*s
qrt(e*x + d))/(b*x + a)) - 2*(B*b^2*d^2 + A*a*b*e^2 - (B*a*b + A*b^2)*d*e)*sqrt(e*x + d))/(b^3*d^3*e - 2*a*b^2
*d^2*e^2 + a^2*b*d*e^3 + (b^3*d^2*e^2 - 2*a*b^2*d*e^3 + a^2*b*e^4)*x), -2*(((B*a - A*b)*e^2*x + (B*a - A*b)*d*
e)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (B*b^2*d^2 + A*a*b*e^2 - (B
*a*b + A*b^2)*d*e)*sqrt(e*x + d))/(b^3*d^3*e - 2*a*b^2*d^2*e^2 + a^2*b*d*e^3 + (b^3*d^2*e^2 - 2*a*b^2*d*e^3 +
a^2*b*e^4)*x)]

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Sympy [A]  time = 15.2266, size = 76, normalized size = 0.86 \begin{align*} \frac{2 \left (- A e + B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{2 \left (- A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}} \left (a e - b d\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.60328, size = 126, normalized size = 1.43 \begin{align*} -\frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} - \frac{2 \,{\left (B d - A e\right )}}{{\left (b d e - a e^{2}\right )} \sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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