\(\int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx\) [383]

Optimal result

Integrand size = 22, antiderivative size = 318 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx=\frac {b \left (945 b^3 c^3-1785 a b^2 c^2 d+839 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x}}{192 a^5 c \sqrt {a+b x}}+\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {5 (b c-a d)^2 \left (63 b^2 c^2-14 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{11/2} c^{3/2}} \] Output:

1/192*b*(-15*a^3*d^3+839*a^2*b*c*d^2-1785*a*b^2*c^2*d+945*b^3*c^3)*(d*x+c) 
^(1/2)/a^5/c/(b*x+a)^(1/2)+1/24*c*(-11*a*d+9*b*c)*(d*x+c)^(1/2)/a^2/x^3/(b 
*x+a)^(1/2)-1/96*(-59*a*d+63*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^3/x^2/(b*x+a) 
^(1/2)+1/192*(-a*d+b*c)*(15*a^2*d^2-322*a*b*c*d+315*b^2*c^2)*(d*x+c)^(1/2) 
/a^4/c/x/(b*x+a)^(1/2)-1/4*c*(d*x+c)^(3/2)/a/x^4/(b*x+a)^(1/2)-5/64*(-a*d+ 
b*c)^2*(-a^2*d^2-14*a*b*c*d+63*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1 
/2)/(d*x+c)^(1/2))/a^(11/2)/c^(3/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (945 b^4 c^3 x^4+105 a b^3 c^2 x^3 (3 c-17 d x)+a^2 b^2 c x^2 \left (-126 c^2-637 c d x+839 d^2 x^2\right )+a^3 b x \left (72 c^3+244 c^2 d x+337 c d^2 x^2-15 d^3 x^3\right )-a^4 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a^5 c x^4 \sqrt {a+b x}}+\frac {5 (b c-a d)^2 \left (-63 b^2 c^2+14 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{11/2} c^{3/2}} \] Input:

Integrate[(c + d*x)^(5/2)/(x^5*(a + b*x)^(3/2)),x]
 

Output:

(Sqrt[c + d*x]*(945*b^4*c^3*x^4 + 105*a*b^3*c^2*x^3*(3*c - 17*d*x) + a^2*b 
^2*c*x^2*(-126*c^2 - 637*c*d*x + 839*d^2*x^2) + a^3*b*x*(72*c^3 + 244*c^2* 
d*x + 337*c*d^2*x^2 - 15*d^3*x^3) - a^4*(48*c^3 + 136*c^2*d*x + 118*c*d^2* 
x^2 + 15*d^3*x^3)))/(192*a^5*c*x^4*Sqrt[a + b*x]) + (5*(b*c - a*d)^2*(-63* 
b^2*c^2 + 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*S 
qrt[a + b*x])])/(64*a^(11/2)*c^(3/2))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {109, 27, 166, 27, 168, 27, 168, 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.

\(\displaystyle \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (9 b c-11 a d)+2 d (3 b c-4 a d) x)}{2 x^4 (a+b x)^{3/2}}dx}{4 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (9 b c-11 a d)+2 d (3 b c-4 a d) x)}{x^4 (a+b x)^{3/2}}dx}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {\frac {\int -\frac {c (63 b c-59 a d) (b c-a d)+6 d (9 b c-8 a d) x (b c-a d)}{2 x^3 (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {c (63 b c-59 a d) (b c-a d)+6 d (9 b c-8 a d) x (b c-a d)}{x^3 (a+b x)^{3/2} \sqrt {c+d x}}dx}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {c (b c-a d) \left (315 b^2 c^2-322 a b d c+15 a^2 d^2+4 b d (63 b c-59 a d) x\right )}{2 x^2 (a+b x)^{3/2} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \int \frac {315 b^2 c^2-322 a b d c+15 a^2 d^2+4 b d (63 b c-59 a d) x}{x^2 (a+b x)^{3/2} \sqrt {c+d x}}dx}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \left (-\frac {\int \frac {15 (b c-a d) \left (63 b^2 c^2-14 a b d c-a^2 d^2\right )+2 b d \left (315 b^2 c^2-322 a b d c+15 a^2 d^2\right ) x}{2 x (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {c+d x} \left (\frac {315 b^2 c}{a}+\frac {15 a d^2}{c}-322 b d\right )}{x \sqrt {a+b x}}\right )}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \left (-\frac {\int \frac {15 (b c-a d) \left (63 b^2 c^2-14 a b d c-a^2 d^2\right )+2 b d \left (315 b^2 c^2-322 a b d c+15 a^2 d^2\right ) x}{x (a+b x)^{3/2} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {315 b^2 c}{a}+\frac {15 a d^2}{c}-322 b d\right )}{x \sqrt {a+b x}}\right )}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \left (-\frac {\frac {2 \int \frac {15 (b c-a d)^2 \left (63 b^2 c^2-14 a b d c-a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {315 b^2 c}{a}+\frac {15 a d^2}{c}-322 b d\right )}{x \sqrt {a+b x}}\right )}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \left (-\frac {\frac {15 (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}+\frac {2 b \sqrt {c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {315 b^2 c}{a}+\frac {15 a d^2}{c}-322 b d\right )}{x \sqrt {a+b x}}\right )}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \left (-\frac {\frac {30 (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}+\frac {2 b \sqrt {c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {315 b^2 c}{a}+\frac {15 a d^2}{c}-322 b d\right )}{x \sqrt {a+b x}}\right )}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {-\frac {(b c-a d) \left (-\frac {\frac {2 b \sqrt {c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {30 (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}}{2 a c}-\frac {\sqrt {c+d x} \left (\frac {315 b^2 c}{a}+\frac {15 a d^2}{c}-322 b d\right )}{x \sqrt {a+b x}}\right )}{4 a}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{2 a x^2 \sqrt {a+b x}}}{6 a}-\frac {c \sqrt {c+d x} (9 b c-11 a d)}{3 a x^3 \sqrt {a+b x}}}{8 a}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}\)

Input:

Int[(c + d*x)^(5/2)/(x^5*(a + b*x)^(3/2)),x]
 

Output:

-1/4*(c*(c + d*x)^(3/2))/(a*x^4*Sqrt[a + b*x]) - (-1/3*(c*(9*b*c - 11*a*d) 
*Sqrt[c + d*x])/(a*x^3*Sqrt[a + b*x]) - (-1/2*((63*b*c - 59*a*d)*(b*c - a* 
d)*Sqrt[c + d*x])/(a*x^2*Sqrt[a + b*x]) - ((b*c - a*d)*(-((((315*b^2*c)/a 
- 322*b*d + (15*a*d^2)/c)*Sqrt[c + d*x])/(x*Sqrt[a + b*x])) - ((2*b*(945*b 
^3*c^3 - 1785*a*b^2*c^2*d + 839*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c + d*x])/( 
a*(b*c - a*d)*Sqrt[a + b*x]) - (30*(b*c - a*d)*(63*b^2*c^2 - 14*a*b*c*d - 
a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2 
)*Sqrt[c]))/(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. \(981\) vs. \(2(274)=548\).

Time = 0.26 (sec) , antiderivative size = 982, normalized size of antiderivative = 3.09

method	result	size
default	\(\frac {\sqrt {x d +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{4} b \,d^{4} x^{5}+180 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} b^{2} c \,d^{3} x^{5}-1350 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{2} d^{2} x^{5}+2100 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{4} c^{3} d \,x^{5}-945 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{5} c^{4} x^{5}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{5} d^{4} x^{4}+180 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{4} b c \,d^{3} x^{4}-1350 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{2} x^{4}+2100 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d \,x^{4}-945 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} x^{4}-30 a^{3} b \,d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+1678 a^{2} b^{2} c \,d^{2} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-3570 a \,b^{3} c^{2} d \,x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+1890 b^{4} c^{3} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-30 a^{4} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+674 a^{3} b c \,d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-1274 a^{2} b^{2} c^{2} d \,x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+630 a \,b^{3} c^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-236 a^{4} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+488 a^{3} b \,c^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-252 a^{2} b^{2} c^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-272 a^{4} c^{2} d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+144 a^{3} b \,c^{3} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-96 a^{4} c^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\right )}{384 a^{5} c \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{4} \sqrt {a c}\, \sqrt {b x +a}}\)	\(982\)

Input:

int((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/384*(d*x+c)^(1/2)*(15*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*d^4*x^5+180*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^2*c*d^3*x^5-1350*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^3*c^2*d^2*x^5+2100*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^4*c^3*d*x^5-945*ln(( 
a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^5*c^4*x^5+15 
*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*d^4*x 
^4+180*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*c*d^3*x^4-1350*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^2*c^2*d^2*x^4+2100*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^3*c^3*d*x^4-945*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^4*c^4*x^4-30*a^3*b*d^3*x^4*((b*x+a) 
*(d*x+c))^(1/2)*(a*c)^(1/2)+1678*a^2*b^2*c*d^2*x^4*((b*x+a)*(d*x+c))^(1/2) 
*(a*c)^(1/2)-3570*a*b^3*c^2*d*x^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+1890 
*b^4*c^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30*a^4*d^3*x^3*((b*x+a)*( 
d*x+c))^(1/2)*(a*c)^(1/2)+674*a^3*b*c*d^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(a*c 
)^(1/2)-1274*a^2*b^2*c^2*d*x^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+630*a*b 
^3*c^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-236*a^4*c*d^2*x^2*((b*x+a)* 
(d*x+c))^(1/2)*(a*c)^(1/2)+488*a^3*b*c^2*d*x^2*((b*x+a)*(d*x+c))^(1/2)*(a* 
c)^(1/2)-252*a^2*b^2*c^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-272*a^...
 

Fricas [A] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 836, normalized size of antiderivative = 2.63 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/768*(15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3*c^2*d^2 - 12*a^3*b 
^2*c*d^3 - a^4*b*d^4)*x^5 + (63*a*b^4*c^4 - 140*a^2*b^3*c^3*d + 90*a^3*b^2 
*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*d^4)*x^4)*sqrt(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^5*c^4 - ( 
945*a*b^4*c^4 - 1785*a^2*b^3*c^3*d + 839*a^3*b^2*c^2*d^2 - 15*a^4*b*c*d^3) 
*x^4 - (315*a^2*b^3*c^4 - 637*a^3*b^2*c^3*d + 337*a^4*b*c^2*d^2 - 15*a^5*c 
*d^3)*x^3 + 2*(63*a^3*b^2*c^4 - 122*a^4*b*c^3*d + 59*a^5*c^2*d^2)*x^2 - 8* 
(9*a^4*b*c^4 - 17*a^5*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b*c^2*x^ 
5 + a^7*c^2*x^4), 1/384*(15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3*c^ 
2*d^2 - 12*a^3*b^2*c*d^3 - a^4*b*d^4)*x^5 + (63*a*b^4*c^4 - 140*a^2*b^3*c^ 
3*d + 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*d^4)*x^4)*sqrt(-a*c)*arcta 
n(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^5*c^4 - (945*a*b^4*c 
^4 - 1785*a^2*b^3*c^3*d + 839*a^3*b^2*c^2*d^2 - 15*a^4*b*c*d^3)*x^4 - (315 
*a^2*b^3*c^4 - 637*a^3*b^2*c^3*d + 337*a^4*b*c^2*d^2 - 15*a^5*c*d^3)*x^3 + 
 2*(63*a^3*b^2*c^4 - 122*a^4*b*c^3*d + 59*a^5*c^2*d^2)*x^2 - 8*(9*a^4*b*c^ 
4 - 17*a^5*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b*c^2*x^5 + a^7*c^2 
*x^4)]
 

Sympy [F]

\[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{5} \left (a + b x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((d*x+c)**(5/2)/x**5/(b*x+a)**(3/2),x)
 

Output:

Integral((c + d*x)**(5/2)/(x**5*(a + b*x)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),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. 3947 vs. \(2 (274) = 548\).

Time = 8.72 (sec) , antiderivative size = 3947, normalized size of antiderivative = 12.41 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),x, algorithm="giac")
 

Output:

4*(sqrt(b*d)*b^4*c^3*abs(b) - 3*sqrt(b*d)*a*b^3*c^2*d*abs(b) + 3*sqrt(b*d) 
*a^2*b^2*c*d^2*abs(b) - sqrt(b*d)*a^3*b*d^3*abs(b))/((b^2*c - a*b*d - (sqr 
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*a^5) - 5/64 
*(63*sqrt(b*d)*b^4*c^4*abs(b) - 140*sqrt(b*d)*a*b^3*c^3*d*abs(b) + 90*sqrt 
(b*d)*a^2*b^2*c^2*d^2*abs(b) - 12*sqrt(b*d)*a^3*b*c*d^3*abs(b) - sqrt(b*d) 
*a^4*d^4*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - s 
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d) 
*a^5*b*c) + 1/96*(561*sqrt(b*d)*b^18*c^11*abs(b) - 5505*sqrt(b*d)*a*b^17*c 
^10*d*abs(b) + 24299*sqrt(b*d)*a^2*b^16*c^9*d^2*abs(b) - 63547*sqrt(b*d)*a 
^3*b^15*c^8*d^3*abs(b) + 109082*sqrt(b*d)*a^4*b^14*c^7*d^4*abs(b) - 128506 
*sqrt(b*d)*a^5*b^13*c^6*d^5*abs(b) + 105350*sqrt(b*d)*a^6*b^12*c^5*d^6*abs 
(b) - 59494*sqrt(b*d)*a^7*b^11*c^4*d^7*abs(b) + 22277*sqrt(b*d)*a^8*b^10*c 
^3*d^8*abs(b) - 5077*sqrt(b*d)*a^9*b^9*c^2*d^9*abs(b) + 575*sqrt(b*d)*a^10 
*b^8*c*d^10*abs(b) - 15*sqrt(b*d)*a^11*b^7*d^11*abs(b) - 3927*sqrt(b*d)*(s 
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^16*c^10* 
abs(b) + 26262*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^15*c^9*d*abs(b) - 70411*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^14*c^8*d^2*abs(b) + 8 
7560*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^13*c^7*d^3*abs(b) - 22494*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^5\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:

int((c + d*x)^(5/2)/(x^5*(a + b*x)^(3/2)),x)
                                                                                    
                                                                                    
 

Output:

int((c + d*x)^(5/2)/(x^5*(a + b*x)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 10.30 (sec) , antiderivative size = 1628, normalized size of antiderivative = 5.12 \[ \int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(5/2)/x^5/(b*x+a)^(3/2),x)
 

Output:

( - 105*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b 
)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a* 
*5*d**5*x**4 - 1395*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sq 
rt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt( 
c + d*x))*a**4*b*c*d**4*x**4 + 7830*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - s 
qrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) 
 + sqrt(b)*sqrt(c + d*x))*a**3*b**2*c**2*d**3*x**4 - 2550*sqrt(c)*sqrt(a)* 
sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + 
 sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**2*b**3*c**3*d**2*x**4 - 
 12285*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b) 
*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b 
**4*c**4*d*x**4 + 8505*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d) 
*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sq 
rt(c + d*x))*b**5*c**5*x**4 - 105*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2 
*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sq 
rt(b)*sqrt(c + d*x))*a**5*d**5*x**4 - 1395*sqrt(c)*sqrt(a)*sqrt(a + b*x)*l 
og(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + 
b*x) + sqrt(b)*sqrt(c + d*x))*a**4*b*c*d**4*x**4 + 7830*sqrt(c)*sqrt(a)*sq 
rt(a + b*x)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt 
(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b**2*c**2*d**3*x**4 - 2...