3.1717 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx\)

Optimal. Leaf size=170 \[ -\frac{\sqrt{b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac{\sqrt{c+d x} (b c-a d)}{b (a+b x) (b e-a f)}+\frac{2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{f} (b e-a f)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.273094, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 156, 63, 208, 205} \[ -\frac{\sqrt{b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac{\sqrt{c+d x} (b c-a d)}{b (a+b x) (b e-a f)}+\frac{2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{f} (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 98

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

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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]

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.654617, size = 265, normalized size = 1.56 \[ \frac{\frac{(-a d f-2 b c f+3 b d e) \left (\sqrt{b} \sqrt{c+d x} (-3 a d+4 b c+b d x)-3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )\right )}{b^{3/2} (b e-a f)}+\frac{2 (b c-a d) \left (\sqrt{f} \sqrt{c+d x} \sqrt{d e-c f} (4 c f-3 d e+d f x)+3 (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )\right )}{\sqrt{f} (b e-a f) \sqrt{d e-c f}}-\frac{3 b (c+d x)^{5/2}}{a+b x}}{3 (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b*(c + d*x)^(5/2))/(a + b*x) + (2*(b*c - a*d)*(Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[c + d*x]*(-3*d*e + 4*c*f + d*
f*x) + 3*(d*e - c*f)^2*ArcTan[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]]))/(Sqrt[f]*(b*e - a*f)*Sqrt[d*e - c*f])
+ ((3*b*d*e - 2*b*c*f - a*d*f)*(Sqrt[b]*Sqrt[c + d*x]*(4*b*c - 3*a*d + b*d*x) - 3*(b*c - a*d)^(3/2)*ArcTanh[(
Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(b^(3/2)*(b*e - a*f)))/(3*(b*c - a*d)*(b*e - a*f))

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 549, normalized size = 3.2 \begin{align*} -{\frac{{a}^{2}{d}^{2}f}{ \left ( af-be \right ) ^{2}b \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{acdf}{ \left ( af-be \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{a{d}^{2}e}{ \left ( af-be \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-{\frac{bcde}{ \left ( af-be \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{{a}^{2}{d}^{2}f}{ \left ( af-be \right ) ^{2}b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{acdf}{ \left ( af-be \right ) ^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-2\,{\frac{b{c}^{2}f}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-3\,{\frac{a{d}^{2}e}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+3\,{\frac{bcde}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{{c}^{2}{f}^{2}}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( cf-de \right ) f}}{\it Artanh} \left ({\frac{\sqrt{dx+c}f}{\sqrt{ \left ( cf-de \right ) f}}} \right ) }+4\,{\frac{cdef}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( cf-de \right ) f}}{\it Artanh} \left ({\frac{\sqrt{dx+c}f}{\sqrt{ \left ( cf-de \right ) f}}} \right ) }-2\,{\frac{{d}^{2}{e}^{2}}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( cf-de \right ) f}}{\it Artanh} \left ({\frac{\sqrt{dx+c}f}{\sqrt{ \left ( cf-de \right ) f}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^2/(a*f-b*e)^2/b*(d*x+c)^(1/2)/(b*d*x+a*d)*a^2*f+d/(a*f-b*e)^2*(d*x+c)^(1/2)/(b*d*x+a*d)*a*f*c+d^2/(a*f-b*e)
^2*(d*x+c)^(1/2)/(b*d*x+a*d)*e*a-d/(a*f-b*e)^2*(d*x+c)^(1/2)/(b*d*x+a*d)*e*b*c+d^2/(a*f-b*e)^2/b/((a*d-b*c)*b)
^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*a^2*f+d/(a*f-b*e)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1
/2)/((a*d-b*c)*b)^(1/2))*a*f*c-2/(a*f-b*e)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*c
^2*f*b-3*d^2/(a*f-b*e)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*e*a+3*d/(a*f-b*e)^2/(
(a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*e*b*c-2/(a*f-b*e)^2/((c*f-d*e)*f)^(1/2)*arctanh
((d*x+c)^(1/2)*f/((c*f-d*e)*f)^(1/2))*c^2*f^2+4*d/(a*f-b*e)^2/((c*f-d*e)*f)^(1/2)*arctanh((d*x+c)^(1/2)*f/((c*
f-d*e)*f)^(1/2))*c*e*f-2*d^2/(a*f-b*e)^2/((c*f-d*e)*f)^(1/2)*arctanh((d*x+c)^(1/2)*f/((c*f-d*e)*f)^(1/2))*e^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((d*x+c)^(3/2)/(b*x+a)^2/(f*x+e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 6.68865, size = 2390, normalized size = 14.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt((b*c - a*d)/b)*log((b*d*x
+ 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)
*x)*sqrt(-(d*e - c*f)/f)*log((d*f*x - d*e + 2*c*f - 2*sqrt(d*x + c)*f*sqrt(-(d*e - c*f)/f))/(f*x + e)) + 2*((b
^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^
3*e*f + a^2*b^2*f^2)*x), -((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt(-(b*c
- a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)
*x)*sqrt(-(d*e - c*f)/f)*log((d*f*x - d*e + 2*c*f - 2*sqrt(d*x + c)*f*sqrt(-(d*e - c*f)/f))/(f*x + e)) + ((b^2
*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*
e*f + a^2*b^2*f^2)*x), -1/2*(4*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)*x)*sqrt((d*e - c*f)/f)*arctan(-sqrt(d*
x + c)*f*sqrt((d*e - c*f)/f)/(d*e - c*f)) + (3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*
f)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*((b
^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^
3*e*f + a^2*b^2*f^2)*x), -((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt(-(b*c
- a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + 2*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*
f)*x)*sqrt((d*e - c*f)/f)*arctan(-sqrt(d*x + c)*f*sqrt((d*e - c*f)/f)/(d*e - c*f)) + ((b^2*c - a*b*d)*e - (a*b
*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f^2)*x
)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.21331, size = 338, normalized size = 1.99 \begin{align*} -\frac{{\left (2 \, b^{2} c^{2} f - a b c d f - a^{2} d^{2} f - 3 \, b^{2} c d e + 3 \, a b d^{2} e\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b f^{2} - 2 \, a b^{2} f e + b^{3} e^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} f}{\sqrt{-c f^{2} + d f e}}\right )}{{\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )} \sqrt{-c f^{2} + d f e}} + \frac{\sqrt{d x + c} b c d - \sqrt{d x + c} a d^{2}}{{\left (a b f - b^{2} e\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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