3.459 \(\int \frac{(c+d x)^{5/2}}{x (a+b x)} \, dx\)

Optimal. Leaf size=118 \[ \frac{2 d \sqrt{c+d x} (2 b c-a d)}{b^2}+\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 d (c+d x)^{3/2}}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.127005, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {84, 154, 156, 63, 208} \[ \frac{2 d \sqrt{c+d x} (2 b c-a d)}{b^2}+\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 d (c+d x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(x*(a + b*x)),x]

[Out]

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

Rule 84

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

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{5/2}}{x (a+b x)} \, dx &=\frac{2 d (c+d x)^{3/2}}{3 b}+\frac{\int \frac{\sqrt{c+d x} \left (b c^2+d (2 b c-a d) x\right )}{x (a+b x)} \, dx}{b}\\ &=\frac{2 d (2 b c-a d) \sqrt{c+d x}}{b^2}+\frac{2 d (c+d x)^{3/2}}{3 b}+\frac{2 \int \frac{\frac{b^2 c^3}{2}+\frac{1}{2} d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{x (a+b x) \sqrt{c+d x}} \, dx}{b^2}\\ &=\frac{2 d (2 b c-a d) \sqrt{c+d x}}{b^2}+\frac{2 d (c+d x)^{3/2}}{3 b}+\frac{c^3 \int \frac{1}{x \sqrt{c+d x}} \, dx}{a}-\frac{(b c-a d)^3 \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{a b^2}\\ &=\frac{2 d (2 b c-a d) \sqrt{c+d x}}{b^2}+\frac{2 d (c+d x)^{3/2}}{3 b}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{a d}-\frac{\left (2 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{a b^2 d}\\ &=\frac{2 d (2 b c-a d) \sqrt{c+d x}}{b^2}+\frac{2 d (c+d x)^{3/2}}{3 b}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.128842, size = 107, normalized size = 0.91 \[ \frac{2 d \sqrt{c+d x} (-3 a d+7 b c+b d x)}{3 b^2}+\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(x*(a + b*x)),x]

[Out]

(2*d*Sqrt[c + d*x]*(7*b*c - 3*a*d + b*d*x))/(3*b^2) - (2*c^(5/2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a + (2*(b*c -
a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*b^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 237, normalized size = 2. \begin{align*}{\frac{2\,d}{3\,b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{a{d}^{2}\sqrt{dx+c}}{{b}^{2}}}+4\,{\frac{d\sqrt{dx+c}c}{b}}-2\,{\frac{{c}^{5/2}}{a}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{a}^{2}{d}^{3}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{ac{d}^{2}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{d{c}^{2}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{b{c}^{3}}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/x/(b*x+a),x)

[Out]

2/3*d*(d*x+c)^(3/2)/b-2/b^2*a*d^2*(d*x+c)^(1/2)+4*d/b*(d*x+c)^(1/2)*c-2*c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1/2))
/a+2/b^2*a^2*d^3/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-6/b*a*d^2/((a*d-b*c)*b)^(1/2)
*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*c+6*d/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1
/2))*c^2-2*b/a/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))*c^3

________________________________________________________________________________________

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)^(5/2)/x/(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 4.02978, size = 1350, normalized size = 11.44 \begin{align*} \left [\frac{3 \, b^{2} c^{\frac{5}{2}} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, \frac{3 \, b^{2} c^{\frac{5}{2}} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, \frac{6 \, b^{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, \frac{2 \,{\left (3 \, b^{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) +{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}\right )}}{3 \, a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x/(b*x+a),x, algorithm="fricas")

[Out]

[1/3*(3*b^2*c^(5/2)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^2*x + 7*a*b
*c*d - 3*a^2*d^2)*sqrt(d*x + c))/(a*b^2), 1/3*(3*b^2*c^(5/2)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 6*
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^2*x + 7*a*b*c*d - 3*a^2*d^2)*sqrt(d*x + c))/(a*b^2), 1/3*(6*b^2*sqrt(-c)*c^2*arctan(sqrt(d*x + c)*
sqrt(-c)/c) + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^2*x + 7*a*b*c*d - 3*a^2*d^2)*sqrt(d*x + c))/(a*b^2), 2/3*(3*b^2*
sqrt(-c)*c^2*arctan(sqrt(d*x + c)*sqrt(-c)/c) + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^2*x + 7*a*b*c*d - 3*a^2*d^2)*sqrt(d*x + c))/(a*b^2
)]

________________________________________________________________________________________

Sympy [A]  time = 72.4441, size = 119, normalized size = 1.01 \begin{align*} \frac{2 d \left (c + d x\right )^{\frac{3}{2}}}{3 b} + \frac{\sqrt{c + d x} \left (- 2 a d^{2} + 4 b c d\right )}{b^{2}} + \frac{2 c^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{a \sqrt{- c}} + \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{a b^{3} \sqrt{\frac{a d - b c}{b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/2)/x/(b*x+a),x)

[Out]

2*d*(c + d*x)**(3/2)/(3*b) + sqrt(c + d*x)*(-2*a*d**2 + 4*b*c*d)/b**2 + 2*c**3*atan(sqrt(c + d*x)/sqrt(-c))/(a
*sqrt(-c)) + 2*(a*d - b*c)**3*atan(sqrt(c + d*x)/sqrt((a*d - b*c)/b))/(a*b**3*sqrt((a*d - b*c)/b))

________________________________________________________________________________________

Giac [A]  time = 1.16669, size = 208, normalized size = 1.76 \begin{align*} \frac{2 \, c^{3} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a \sqrt{-c}} - \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a b^{2}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{2} d + 6 \, \sqrt{d x + c} b^{2} c d - 3 \, \sqrt{d x + c} a b d^{2}\right )}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x/(b*x+a),x, algorithm="giac")

[Out]

2*c^3*arctan(sqrt(d*x + c)/sqrt(-c))/(a*sqrt(-c)) - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arct
an(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a*b^2) + 2/3*((d*x + c)^(3/2)*b^2*d + 6*sqrt(d*
x + c)*b^2*c*d - 3*sqrt(d*x + c)*a*b*d^2)/b^3