\(\int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx\) [296]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {a^{3/2} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \left (-a^2 d+b^2 c x\right ) \sqrt {c+d x}}{c d x}+\frac {a^{3/2} (-5 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {109, 27, 171, 175, 66, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 175

\(\displaystyle \frac {\frac {a^2 d (5 b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-b^2 c (b c-5 a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{d}+\frac {2 b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{d}}{2 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {a^2 d (5 b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 b^2 c (b c-5 a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}+\frac {2 b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{d}}{2 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {2 a^2 d (5 b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-2 b^2 c (b c-5 a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}+\frac {2 b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{d}}{2 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {2 a^{3/2} d (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {2 b^{3/2} c (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}}{d}+\frac {2 b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{d}}{2 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}\)

Input:

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

Output:

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

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[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] /; Fre 
eQ[{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 175

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

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. \(319\) vs. \(2(130)=260\).

Time = 0.23 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.98

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (5 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c d x \sqrt {a c}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{3} c^{2} x \sqrt {a c}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x \sqrt {d b}-5 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b c d x \sqrt {d b}+2 b^{2} c x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}-2 a^{2} d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\right )}{2 c \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {d b}\, \sqrt {a c}\, d}\)	\(320\)

Input:

int((b*x+a)^(5/2)/x^2/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 1.10 (sec) , antiderivative size = 993, normalized size of antiderivative = 6.13 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)/x^2/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

[-1/4*((b^2*c^2 - 5*a*b*c*d)*x*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a 
*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + 
c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + (5*a*b*c*d - a^2*d^2)*x*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^2 + (b*c 
^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c* 
d)*x)/x^2) - 4*(b^2*c*x - a^2*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d*x), 1/4 
*(2*(b^2*c^2 - 5*a*b*c*d)*x*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sq 
rt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)* 
x)) - (5*a*b*c*d - a^2*d^2)*x*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d* 
x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(b^2*c*x - a^2*d)*sqr 
t(b*x + a)*sqrt(d*x + c))/(c*d*x), 1/4*(2*(5*a*b*c*d - a^2*d^2)*x*sqrt(-a/ 
c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-a/ 
c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - (b^2*c^2 - 5*a*b*c*d)*x*sqrt 
(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b 
*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2 
)*x) + 4*(b^2*c*x - a^2*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d*x), 1/2*((5*a 
*b*c*d - a^2*d^2)*x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x 
 + a)*sqrt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) + 
(b^2*c^2 - 5*a*b*c*d)*x*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqr...
 

Sympy [F]

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

integrate((b*x+a)**(5/2)/x**2/(d*x+c)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(5/2)/x^2/(d*x+c)^(1/2),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. 517 vs. \(2 (130) = 260\).

Time = 0.28 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.19 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {b^{3} {\left (\frac {{\left (b c - 5 \, a d\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d} d} + \frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}}{b d} - \frac {2 \, {\left (5 \, \sqrt {b d} a^{2} b c - \sqrt {b d} a^{3} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b^{2} c} - \frac {4 \, {\left (\sqrt {b d} a^{2} b^{3} c^{2} - 2 \, \sqrt {b d} a^{3} b^{2} c d + \sqrt {b d} a^{4} b d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} b c}\right )}}{2 \, {\left | b \right |}} \] Input:

integrate((b*x+a)^(5/2)/x^2/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

1/2*b^3*((b*c - 5*a*d)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + 
a)*b*d - a*b*d))^2)/(sqrt(b*d)*d) + 2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)* 
sqrt(b*x + a)/(b*d) - 2*(5*sqrt(b*d)*a^2*b*c - sqrt(b*d)*a^3*d)*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)*b^2*c) - 4*(sqrt(b*d)*a^2* 
b^3*c^2 - 2*sqrt(b*d)*a^3*b^2*c*d + sqrt(b*d)*a^4*b*d^2 - 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*c - 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*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - 
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) 
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) 
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*b*c))/abs(b)
 

Mupad [F(-1)]

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

int((a + b*x)^(5/2)/(x^2*(c + d*x)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*x)^(5/2)/(x^2*(c + d*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.77 \[ \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} c \,d^{2}+2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{2} d x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a^{2} d^{3} x +5 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b c \,d^{2} x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a^{2} d^{3} x +5 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b c \,d^{2} x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a^{2} d^{3} x -5 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a b c \,d^{2} x +10 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b \,c^{2} d x -2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{3} x}{2 c^{2} d^{2} x} \] Input:

int((b*x+a)^(5/2)/x^2/(d*x+c)^(1/2),x)
 

Output:

( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a**2*c*d**2 + 2*sqrt(c + d*x)*sqrt(a + b 
*x)*b**2*c**2*d*x - sqrt(c)*sqrt(a)*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 
*d**3*x + 5*sqrt(c)*sqrt(a)*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*c*d**2*x 
 - sqrt(c)*sqrt(a)*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*d**3*x + 5*sqrt(c)* 
sqrt(a)*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*c*d**2*x + sqrt(c)*sqrt(a)*log( 
2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)* 
sqrt(a) + 2*b*d*x)*a**2*d**3*x - 5*sqrt(c)*sqrt(a)*log(2*sqrt(d)*sqrt(b)*s 
qrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)* 
a*b*c*d**2*x + 10*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqr 
t(c + d*x))/sqrt(a*d - b*c))*a*b*c**2*d*x - 2*sqrt(d)*sqrt(b)*log((sqrt(d) 
*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**3*x)/(2*c 
**2*d**2*x)