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

Optimal. Leaf size=128 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{d \sqrt{c+d x} (2 a d+b c)}{a b}-\frac{c (c+d x)^{3/2}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.168315, antiderivative size = 128, 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 = {98, 154, 156, 63, 208} \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{d \sqrt{c+d x} (2 a d+b c)}{a b}-\frac{c (c+d x)^{3/2}}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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^2 (a+b x)} \, dx &=-\frac{c (c+d x)^{3/2}}{a x}-\frac{\int \frac{\sqrt{c+d x} \left (\frac{1}{2} c (2 b c-5 a d)-\frac{1}{2} d (b c+2 a d) x\right )}{x (a+b x)} \, dx}{a}\\ &=\frac{d (b c+2 a d) \sqrt{c+d x}}{a b}-\frac{c (c+d x)^{3/2}}{a x}-\frac{2 \int \frac{\frac{1}{4} b c^2 (2 b c-5 a d)+\frac{1}{4} d \left (b^2 c^2-6 a b c d+2 a^2 d^2\right ) x}{x (a+b x) \sqrt{c+d x}} \, dx}{a b}\\ &=\frac{d (b c+2 a d) \sqrt{c+d x}}{a b}-\frac{c (c+d x)^{3/2}}{a x}-\frac{\left (c^2 (2 b c-5 a d)\right ) \int \frac{1}{x \sqrt{c+d x}} \, dx}{2 a^2}+\frac{(b c-a d)^3 \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{a^2 b}\\ &=\frac{d (b c+2 a d) \sqrt{c+d x}}{a b}-\frac{c (c+d x)^{3/2}}{a x}-\frac{\left (c^2 (2 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{a^2 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^2 b d}\\ &=\frac{d (b c+2 a d) \sqrt{c+d x}}{a b}-\frac{c (c+d x)^{3/2}}{a x}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^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^2 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.119664, size = 114, normalized size = 0.89 \[ \frac{-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}}+\frac{a \sqrt{c+d x} \left (2 a d^2 x-b c^2\right )}{b x}+c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.016, size = 249, normalized size = 2. \begin{align*} 2\,{\frac{{d}^{2}\sqrt{dx+c}}{b}}-{\frac{{c}^{2}}{ax}\sqrt{dx+c}}-5\,{\frac{d{c}^{3/2}}{a}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{c}^{5/2}b}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{{d}^{3}a}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{d}^{2}c}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{bd{c}^{2}}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{b}^{2}{c}^{3}}{{a}^{2}\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^2/(b*x+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.04736, size = 1392, normalized size = 10.88 \begin{align*} \left [\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \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 ) -{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{c} x \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{2 \, a^{2} b x}, -\frac{4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \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 (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{c} x \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{2 \, a^{2} b x}, -\frac{{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \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 ) -{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{a^{2} b x}, -\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \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 (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) -{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{a^{2} b x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqr
t((b*c - a*d)/b))/(b*x + a)) - (2*b^2*c^2 - 5*a*b*c*d)*sqrt(c)*x*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x)
+ 2*(2*a^2*d^2*x - a*b*c^2)*sqrt(d*x + c))/(a^2*b*x), -1/2*(4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt(-(b*c - a
*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (2*b^2*c^2 - 5*a*b*c*d)*sqrt(c)*x*log((d*x
- 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(2*a^2*d^2*x - a*b*c^2)*sqrt(d*x + c))/(a^2*b*x), -((2*b^2*c^2 - 5*a*b
*c*d)*sqrt(-c)*x*arctan(sqrt(d*x + c)*sqrt(-c)/c) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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^2*d^2*x - a*b*c^2)*sqrt(d*x +
c))/(a^2*b*x), -(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c -
a*d)/b)/(b*c - a*d)) + (2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)*x*arctan(sqrt(d*x + c)*sqrt(-c)/c) - (2*a^2*d^2*x - a
*b*c^2)*sqrt(d*x + c))/(a^2*b*x)]

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Sympy [B]  time = 87.4956, size = 333, normalized size = 2.6 \begin{align*} - \frac{2 a d^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d}{b} - c}} \right )}}{b^{2} \sqrt{\frac{a d}{b} - c}} + \frac{6 c d^{2} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d}{b} - c}} \right )}}{b \sqrt{\frac{a d}{b} - c}} + \frac{2 d^{2} \sqrt{c + d x}}{b} - \frac{c^{3} d \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )}}{2 a} + \frac{c^{3} d \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )}}{2 a} - \frac{6 c^{2} d \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d}{b} - c}} \right )}}{a \sqrt{\frac{a d}{b} - c}} + \frac{6 c^{2} d \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{a \sqrt{- c}} - \frac{c^{2} \sqrt{c + d x}}{a x} + \frac{2 b c^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d}{b} - c}} \right )}}{a^{2} \sqrt{\frac{a d}{b} - c}} - \frac{2 b c^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{a^{2} \sqrt{- c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*d**3*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b**2*sqrt(a*d/b - c)) + 6*c*d**2*atan(sqrt(c + d*x)/sqrt(a*d/b
- c))/(b*sqrt(a*d/b - c)) + 2*d**2*sqrt(c + d*x)/b - c**3*d*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d
*x))/(2*a) + c**3*d*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2*a) - 6*c**2*d*atan(sqrt(c + d*x)/
sqrt(a*d/b - c))/(a*sqrt(a*d/b - c)) + 6*c**2*d*atan(sqrt(c + d*x)/sqrt(-c))/(a*sqrt(-c)) - c**2*sqrt(c + d*x)
/(a*x) + 2*b*c**3*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(a**2*sqrt(a*d/b - c)) - 2*b*c**3*atan(sqrt(c + d*x)/sqr
t(-c))/(a**2*sqrt(-c))

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Giac [A]  time = 1.2078, size = 205, normalized size = 1.6 \begin{align*} \frac{2 \, \sqrt{d x + c} d^{2}}{b} - \frac{\sqrt{d x + c} c^{2}}{a x} - \frac{{\left (2 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{2} \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^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(d*x + c)*d^2/b - sqrt(d*x + c)*c^2/(a*x) - (2*b*c^3 - 5*a*c^2*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^2*sq
rt(-c)) + 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(
sqrt(-b^2*c + a*b*d)*a^2*b)