3.1770 \(\int \frac{a+b x}{(c+d x) (e+f x)^{3/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0797027, 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} \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{3/2}}-\frac{2 (b e-a f)}{f \sqrt{e+f x} (d e-c f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*e - a*f))/(f*(d*e - c*f)*Sqrt[e + f*x]) + (2*(b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f
]])/(Sqrt[d]*(d*e - c*f)^(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}{(c+d x) (e+f x)^{3/2}} \, dx &=-\frac{2 (b e-a f)}{f (d e-c f) \sqrt{e+f x}}-\frac{(b c-a d) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d e-c f}\\ &=-\frac{2 (b e-a f)}{f (d e-c f) \sqrt{e+f x}}-\frac{(2 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{c-\frac{d e}{f}+\frac{d x^2}{f}} \, dx,x,\sqrt{e+f x}\right )}{f (d e-c f)}\\ &=-\frac{2 (b e-a f)}{f (d e-c f) \sqrt{e+f x}}+\frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d} (d e-c f)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.181204, size = 97, normalized size = 1.1 \[ \frac{2 \left (\frac{(b e-a f) (c f-d e)}{\sqrt{e+f x}}+\frac{f (b c-a d) \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{\sqrt{d}}\right )}{f (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 142, normalized size = 1.6 \begin{align*} -2\,{\frac{a}{ \left ( cf-de \right ) \sqrt{fx+e}}}+2\,{\frac{be}{ \left ( cf-de \right ) f\sqrt{fx+e}}}-2\,{\frac{ad}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{bc}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.40139, size = 767, normalized size = 8.72 \begin{align*} \left [\frac{{\left ({\left (b c - a d\right )} f^{2} x +{\left (b c - a d\right )} e f\right )} \sqrt{d^{2} e - c d f} \log \left (\frac{d f x + 2 \, d e - c f + 2 \, \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (b d^{2} e^{2} + a c d f^{2} -{\left (b c d + a d^{2}\right )} e f\right )} \sqrt{f x + e}}{d^{3} e^{3} f - 2 \, c d^{2} e^{2} f^{2} + c^{2} d e f^{3} +{\left (d^{3} e^{2} f^{2} - 2 \, c d^{2} e f^{3} + c^{2} d f^{4}\right )} x}, -\frac{2 \,{\left ({\left ({\left (b c - a d\right )} f^{2} x +{\left (b c - a d\right )} e f\right )} \sqrt{-d^{2} e + c d f} \arctan \left (\frac{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}{d f x + d e}\right ) +{\left (b d^{2} e^{2} + a c d f^{2} -{\left (b c d + a d^{2}\right )} e f\right )} \sqrt{f x + e}\right )}}{d^{3} e^{3} f - 2 \, c d^{2} e^{2} f^{2} + c^{2} d e f^{3} +{\left (d^{3} e^{2} f^{2} - 2 \, c d^{2} e f^{3} + c^{2} d f^{4}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 14.2483, size = 78, normalized size = 0.89 \begin{align*} - \frac{2 \left (a f - b e\right )}{f \sqrt{e + f x} \left (c f - d e\right )} - \frac{2 \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}} \left (c f - d e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*f - b*e)/(f*sqrt(e + f*x)*(c*f - d*e)) - 2*(a*d - b*c)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d*sqrt((
c*f - d*e)/d)*(c*f - d*e))

________________________________________________________________________________________

Giac [A]  time = 2.24186, size = 127, normalized size = 1.44 \begin{align*} \frac{2 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e}{\left (c f - d e\right )}} - \frac{2 \,{\left (a f - b e\right )}}{{\left (c f^{2} - d f e\right )} \sqrt{f x + e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*c - a*d)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*(c*f - d*e)) - 2*(a*f - b*e)/((
c*f^2 - d*f*e)*sqrt(f*x + e))