\(\int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx\) [409]

Optimal result

Integrand size = 22, antiderivative size = 388 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\frac {b (b c-a d) \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) \sqrt {c+d x}}{64 a^5 c (a+b x)^{3/2}}+\frac {11 c (b c-a d) \sqrt {c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {(99 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac {(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt {c+d x}}{64 a^4 c x (a+b x)^{3/2}}+\frac {b \left (1155 b^3 c^3-1715 a b^2 c^2 d+581 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {c+d x}}{64 a^6 c \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac {5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{13/2} c^{3/2}} \] Output:

1/64*b*(-a*d+b*c)*(5*a^2*d^2-238*a*b*c*d+385*b^2*c^2)*(d*x+c)^(1/2)/a^5/c/ 
(b*x+a)^(3/2)+11/24*c*(-a*d+b*c)*(d*x+c)^(1/2)/a^2/x^3/(b*x+a)^(3/2)-1/96* 
(-59*a*d+99*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^3/x^2/(b*x+a)^(3/2)+1/64*(-a*d 
+b*c)*(5*a^2*d^2-156*a*b*c*d+231*b^2*c^2)*(d*x+c)^(1/2)/a^4/c/x/(b*x+a)^(3 
/2)+1/64*b*(-5*a^3*d^3+581*a^2*b*c*d^2-1715*a*b^2*c^2*d+1155*b^3*c^3)*(d*x 
+c)^(1/2)/a^6/c/(b*x+a)^(1/2)-1/4*c*(d*x+c)^(3/2)/a/x^4/(b*x+a)^(3/2)-5/64 
*(-a*d+b*c)*(a^3*d^3+21*a^2*b*c*d^2-189*a*b^2*c^2*d+231*b^3*c^3)*arctanh(c 
^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(13/2)/c^(3/2)
 

Mathematica [A] (verified)

Time = 10.36 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\frac {-48 a^{11/2} c^2 (c+d x)^{7/2}+8 a^{9/2} c (11 b c+a d) x (c+d x)^{7/2}-2 a^{7/2} \left (99 b^2 c^2-26 a b c d-a^2 d^2\right ) x^2 (c+d x)^{7/2}+\left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) x^3 \left (3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{192 a^{13/2} c^3 x^4 (a+b x)^{3/2}} \] Input:

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

Output:

(-48*a^(11/2)*c^2*(c + d*x)^(7/2) + 8*a^(9/2)*c*(11*b*c + a*d)*x*(c + d*x) 
^(7/2) - 2*a^(7/2)*(99*b^2*c^2 - 26*a*b*c*d - a^2*d^2)*x^2*(c + d*x)^(7/2) 
 + (231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*x^3*(3*a^(5/ 
2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*(4*a*c + 3*b*c 
*x + a*d*x) - 3*c^(3/2)*(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S 
qrt[a]*Sqrt[c + d*x])])))/(192*a^(13/2)*c^3*x^4*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 169, 27, 169, 27, 104, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {\int \frac {(b c-a d) \sqrt {c+d x} (11 c+8 d x)}{2 x^4 (a+b x)^{5/2}}dx}{4 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \int \frac {\sqrt {c+d x} (11 c+8 d x)}{x^4 (a+b x)^{5/2}}dx}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {(b c-a d) \left (\frac {\int -\frac {c (99 b c-59 a d)+8 d (11 b c-6 a d) x}{2 x^3 (a+b x)^{5/2} \sqrt {c+d x}}dx}{3 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {\int \frac {c (99 b c-59 a d)+8 d (11 b c-6 a d) x}{x^3 (a+b x)^{5/2} \sqrt {c+d x}}dx}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\int \frac {3 c \left (231 b^2 c^2-156 a b d c+5 a^2 d^2+2 b d (99 b c-59 a d) x\right )}{2 x^2 (a+b x)^{5/2} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \int \frac {231 b^2 c^2-156 a b d c+5 a^2 d^2+2 b d (99 b c-59 a d) x}{x^2 (a+b x)^{5/2} \sqrt {c+d x}}dx}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\int \frac {5 \left (231 b^3 c^3-189 a b^2 d c^2+21 a^2 b d^2 c+a^3 d^3\right )+4 b d \left (231 b^2 c^2-156 a b d c+5 a^2 d^2\right ) x}{2 x (a+b x)^{5/2} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\int \frac {5 \left (231 b^3 c^3-189 a b^2 d c^2+21 a^2 b d^2 c+a^3 d^3\right )+4 b d \left (231 b^2 c^2-156 a b d c+5 a^2 d^2\right ) x}{x (a+b x)^{5/2} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\frac {2 \int \frac {3 (b c-a d) \left (5 \left (231 b^3 c^3-189 a b^2 d c^2+21 a^2 b d^2 c+a^3 d^3\right )+2 b d \left (385 b^2 c^2-238 a b d c+5 a^2 d^2\right ) x\right )}{2 x (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a (b c-a d)}+\frac {2 b \sqrt {c+d x} \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{a (a+b x)^{3/2}}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\frac {\int \frac {5 \left (231 b^3 c^3-189 a b^2 d c^2+21 a^2 b d^2 c+a^3 d^3\right )+2 b d \left (385 b^2 c^2-238 a b d c+5 a^2 d^2\right ) x}{x (a+b x)^{3/2} \sqrt {c+d x}}dx}{a}+\frac {2 b \sqrt {c+d x} \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{a (a+b x)^{3/2}}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\frac {\frac {2 \int \frac {5 (b c-a d) \left (231 b^3 c^3-189 a b^2 d c^2+21 a^2 b d^2 c+a^3 d^3\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}}{a}+\frac {2 b \sqrt {c+d x} \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{a (a+b x)^{3/2}}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\frac {\frac {5 \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}+\frac {2 b \sqrt {c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}}{a}+\frac {2 b \sqrt {c+d x} \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{a (a+b x)^{3/2}}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\frac {\frac {10 \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}+\frac {2 b \sqrt {c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}}{a}+\frac {2 b \sqrt {c+d x} \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{a (a+b x)^{3/2}}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {3 \left (-\frac {\frac {2 b \sqrt {c+d x} \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{a (a+b x)^{3/2}}+\frac {\frac {2 b \sqrt {c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {10 \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}}{a}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {231 b^2 c}{a}+\frac {5 a d^2}{c}-156 b d\right )}{x (a+b x)^{3/2}}\right )}{4 a}-\frac {\sqrt {c+d x} (99 b c-59 a d)}{2 a x^2 (a+b x)^{3/2}}}{6 a}-\frac {11 c \sqrt {c+d x}}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}\)

Input:

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

Output:

-1/4*(c*(c + d*x)^(3/2))/(a*x^4*(a + b*x)^(3/2)) - ((b*c - a*d)*((-11*c*Sq 
rt[c + d*x])/(3*a*x^3*(a + b*x)^(3/2)) - (-1/2*((99*b*c - 59*a*d)*Sqrt[c + 
 d*x])/(a*x^2*(a + b*x)^(3/2)) - (3*(-((((231*b^2*c)/a - 156*b*d + (5*a*d^ 
2)/c)*Sqrt[c + d*x])/(x*(a + b*x)^(3/2))) - ((2*b*(385*b^2*c^2 - 238*a*b*c 
*d + 5*a^2*d^2)*Sqrt[c + d*x])/(a*(a + b*x)^(3/2)) + ((2*b*(1155*b^3*c^3 - 
 1715*a*b^2*c^2*d + 581*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[c + d*x])/(a*(b*c - 
a*d)*Sqrt[a + b*x]) - (10*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 
+ a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3 
/2)*Sqrt[c]))/a)/(2*a*c)))/(4*a))/(6*a)))/(8*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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 166

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 221

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1376\) vs. \(2(338)=676\).

Time = 0.27 (sec) , antiderivative size = 1377, normalized size of antiderivative = 3.55

method	result	size
default	\(\text {Expression too large to display}\)	\(1377\)

Input:

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

Output:

1/384*(d*x+c)^(1/2)*(1386*a^2*b^3*c^3*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 
/2)-236*a^5*c*d^2*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-396*a^3*b^2*c^3* 
x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-272*a^5*c^2*d*x*(a*c)^(1/2)*((b*x+ 
a)*(d*x+c))^(1/2)+176*a^4*b*c^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+300* 
ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^3*c* 
d^3*x^6-3150*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/ 
x)*a^2*b^4*c^2*d^2*x^6+6300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c) 
)^(1/2)+2*a*c)/x)*a*b^5*c^3*d*x^6+600*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+ 
a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b^2*c*d^3*x^5-6300*ln((a*d*x+b*c*x+2*(a*c) 
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^3*c^2*d^2*x^5+12600*ln((a*d 
*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^4*c^3*d*x^5 
+300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*b 
*c*d^3*x^4-3150*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a* 
c)/x)*a^4*b^2*c^2*d^2*x^4+6300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x 
+c))^(1/2)+2*a*c)/x)*a^3*b^3*c^3*d*x^4-2322*a^3*b^2*c^2*d*x^3*(a*c)^(1/2)* 
((b*x+a)*(d*x+c))^(1/2)+632*a^4*b*c^2*d*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^ 
(1/2)+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a 
^5*b*d^4*x^5-6930*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2* 
a*c)/x)*a*b^5*c^4*x^5-3465*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c)) 
^(1/2)+2*a*c)/x)*a^2*b^4*c^4*x^4+6930*b^5*c^3*x^5*(a*c)^(1/2)*((b*x+a)*...
 

Fricas [A] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.85 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/768*(15*((231*b^6*c^4 - 420*a*b^5*c^3*d + 210*a^2*b^4*c^2*d^2 - 20*a^3 
*b^3*c*d^3 - a^4*b^2*d^4)*x^6 + 2*(231*a*b^5*c^4 - 420*a^2*b^4*c^3*d + 210 
*a^3*b^3*c^2*d^2 - 20*a^4*b^2*c*d^3 - a^5*b*d^4)*x^5 + (231*a^2*b^4*c^4 - 
420*a^3*b^3*c^3*d + 210*a^4*b^2*c^2*d^2 - 20*a^5*b*c*d^3 - a^6*d^4)*x^4)*s 
qrt(a*c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + 
 (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c 
*d)*x)/x^2) + 4*(48*a^6*c^4 - 3*(1155*a*b^5*c^4 - 1715*a^2*b^4*c^3*d + 581 
*a^3*b^3*c^2*d^2 - 5*a^4*b^2*c*d^3)*x^5 - 6*(770*a^2*b^4*c^4 - 1169*a^3*b^ 
3*c^3*d + 412*a^4*b^2*c^2*d^2 - 5*a^5*b*c*d^3)*x^4 - 3*(231*a^3*b^3*c^4 - 
387*a^4*b^2*c^3*d + 161*a^5*b*c^2*d^2 - 5*a^6*c*d^3)*x^3 + 2*(99*a^4*b^2*c 
^4 - 158*a^5*b*c^3*d + 59*a^6*c^2*d^2)*x^2 - 8*(11*a^5*b*c^4 - 17*a^6*c^3* 
d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^7*b^2*c^2*x^6 + 2*a^8*b*c^2*x^5 + a^ 
9*c^2*x^4), 1/384*(15*((231*b^6*c^4 - 420*a*b^5*c^3*d + 210*a^2*b^4*c^2*d^ 
2 - 20*a^3*b^3*c*d^3 - a^4*b^2*d^4)*x^6 + 2*(231*a*b^5*c^4 - 420*a^2*b^4*c 
^3*d + 210*a^3*b^3*c^2*d^2 - 20*a^4*b^2*c*d^3 - a^5*b*d^4)*x^5 + (231*a^2* 
b^4*c^4 - 420*a^3*b^3*c^3*d + 210*a^4*b^2*c^2*d^2 - 20*a^5*b*c*d^3 - a^6*d 
^4)*x^4)*sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x 
 + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*( 
48*a^6*c^4 - 3*(1155*a*b^5*c^4 - 1715*a^2*b^4*c^3*d + 581*a^3*b^3*c^2*d^2 
- 5*a^4*b^2*c*d^3)*x^5 - 6*(770*a^2*b^4*c^4 - 1169*a^3*b^3*c^3*d + 412*...
 

Sympy [F]

\[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{5} \left (a + b x\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4500 vs. \(2 (338) = 676\).

Time = 3.06 (sec) , antiderivative size = 4500, normalized size of antiderivative = 11.60 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-5/64*(231*sqrt(b*d)*b^4*c^4*abs(b) - 420*sqrt(b*d)*a*b^3*c^3*d*abs(b) + 2 
10*sqrt(b*d)*a^2*b^2*c^2*d^2*abs(b) - 20*sqrt(b*d)*a^3*b*c*d^3*abs(b) - sq 
rt(b*d)*a^4*d^4*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + 
 a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a 
*b*c*d)*a^6*b*c) + 4/3*(15*sqrt(b*d)*b^8*c^5*abs(b) - 68*sqrt(b*d)*a*b^7*c 
^4*d*abs(b) + 122*sqrt(b*d)*a^2*b^6*c^3*d^2*abs(b) - 108*sqrt(b*d)*a^3*b^5 
*c^2*d^3*abs(b) + 47*sqrt(b*d)*a^4*b^4*c*d^4*abs(b) - 8*sqrt(b*d)*a^5*b^3* 
d^5*abs(b) - 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a 
)*b*d - a*b*d))^2*b^6*c^4*abs(b) + 108*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) 
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5*c^3*d*abs(b) - 144*sqrt(b*d 
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^ 
4*c^2*d^2*abs(b) + 84*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d))^2*a^3*b^3*c*d^3*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqr 
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^2*d^4*abs(b) + 1 
5*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d) 
)^4*b^4*c^3*abs(b) - 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d))^4*a*b^3*c^2*d*abs(b) + 27*sqrt(b*d)*(sqrt(b*d)*sqr 
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^2*c*d^2*abs(b) - 
 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d 
))^4*a^3*b*d^3*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqr...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^5\,{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{2}}}{x^{5} \left (b x +a \right )^{\frac {5}{2}}}d x \] Input:

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

Output:

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