\(\int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx\) [239]

Optimal result

Integrand size = 22, antiderivative size = 316 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\frac {\left (3 b^3 c^3-17 a b^2 c^2 d-55 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a^2 c x}-\frac {\left (\frac {3 b^2 c}{a}+50 b d-\frac {5 a d^2}{c}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 x^2}-\frac {(3 b c+5 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 c x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac {\left (3 b^4 c^4-20 a b^3 c^3 d+90 a^2 b^2 c^2 d^2+60 a^3 b c d^3-5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \] Output:

1/64*(5*a^3*d^3-55*a^2*b*c*d^2-17*a*b^2*c^2*d+3*b^3*c^3)*(b*x+a)^(1/2)*(d* 
x+c)^(1/2)/a^2/c/x-1/96*(3*b^2*c/a+50*b*d-5*a*d^2/c)*(b*x+a)^(1/2)*(d*x+c) 
^(3/2)/x^2-1/24*(5*a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(5/2)/c/x^3-1/4*(b*x+a 
)^(3/2)*(d*x+c)^(5/2)/x^4-1/64*(-5*a^4*d^4+60*a^3*b*c*d^3+90*a^2*b^2*c^2*d 
^2-20*a*b^3*c^3*d+3*b^4*c^4)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c) 
^(1/2))/a^(5/2)/c^(3/2)+2*b^(3/2)*d^(5/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^ 
(1/2)/(d*x+c)^(1/2))
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-9 b^3 c^3 x^3+3 a b^2 c^2 x^2 (2 c+19 d x)+a^2 b c x \left (72 c^2+244 c d x+337 d^2 x^2\right )+a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a^2 c x^4}+\frac {\left (-3 b^4 c^4+20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-60 a^3 b c d^3+5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \] Input:

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

Output:

-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*b^3*c^3*x^3 + 3*a*b^2*c^2*x^2*(2*c 
 + 19*d*x) + a^2*b*c*x*(72*c^2 + 244*c*d*x + 337*d^2*x^2) + a^3*(48*c^3 + 
136*c^2*d*x + 118*c*d^2*x^2 + 15*d^3*x^3)))/(a^2*c*x^4) + ((-3*b^4*c^4 + 2 
0*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 5*a^4*d^4)*ArcTanh[( 
Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(64*a^(5/2)*c^(3/2)) + 2* 
b^(3/2)*d^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])]
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 345, 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 = {108, 27, 166, 27, 166, 27, 166, 27, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{4} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} (3 b c+5 a d+8 b d x)}{2 x^4}dx-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} (3 b c+5 a d+8 b d x)}{x^4}dx-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{8} \left (\frac {\int \frac {(c+d x)^{3/2} \left (3 c^2 b^2+48 c d x b^2+50 a c d b-5 a^2 d^2\right )}{2 x^3 \sqrt {a+b x}}dx}{3 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {\int \frac {(c+d x)^{3/2} \left (3 c^2 b^2+48 c d x b^2+50 a c d b-5 a^2 d^2\right )}{x^3 \sqrt {a+b x}}dx}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{8} \left (\frac {\frac {\int -\frac {3 \sqrt {c+d x} \left (3 b^3 c^3-17 a b^2 d c^2-55 a^2 b d^2 c-64 a b^2 d^2 x c+5 a^3 d^3\right )}{2 x^2 \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \int \frac {\sqrt {c+d x} \left (3 b^3 c^3-17 a b^2 d c^2-55 a^2 b d^2 c-64 a b^2 d^2 x c+5 a^3 d^3\right )}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (\frac {\int -\frac {3 b^4 c^4-20 a b^3 d c^3+90 a^2 b^2 d^2 c^2+60 a^3 b d^3 c+128 a^2 b^2 d^3 x c-5 a^4 d^4}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {\int \frac {3 b^4 c^4-20 a b^3 d c^3+90 a^2 b^2 d^2 c^2+60 a^3 b d^3 c+128 a^2 b^2 d^3 x c-5 a^4 d^4}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {128 a^2 b^2 c d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {256 a^2 b^2 c d^3 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {3 \left (-\frac {256 a^2 b^2 c d^3 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 \left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{8} \left (\frac {-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-5 a^2 d^2+50 a b c d+3 b^2 c^2\right )}{2 a x^2}-\frac {3 \left (-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{a x}-\frac {256 a^2 b^{3/2} c d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {2 \left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}}{2 a}\right )}{4 a}}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{3 c x^3}\right )-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}\)

Input:

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

Output:

-1/4*((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^4 + (-1/3*((3*b*c + 5*a*d)*Sqrt[a 
 + b*x]*(c + d*x)^(5/2))/(c*x^3) + (-1/2*((3*b^2*c^2 + 50*a*b*c*d - 5*a^2* 
d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(a*x^2) - (3*(-(((3*b^3*c^3 - 17*a*b^2 
*c^2*d - 55*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - 
 ((-2*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 
5*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt 
[a]*Sqrt[c]) + 256*a^2*b^(3/2)*c*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/( 
Sqrt[b]*Sqrt[c + d*x])])/(2*a)))/(4*a))/(6*c))/8
 

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 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 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. \(732\) vs. \(2(266)=532\).

Time = 0.22 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.32

method	result	size
default	\(\frac {\sqrt {b x +a}\, \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} d^{4} x^{4} \sqrt {d b}-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 c \,d^{3} x^{4} \sqrt {d b}-270 \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^{2} c^{2} d^{2} x^{4} \sqrt {d b}+60 \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^{3} c^{3} d \,x^{4} \sqrt {d b}-9 \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^{4} c^{4} x^{4} \sqrt {d b}+384 \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^{2} b^{2} c \,d^{3} x^{4} \sqrt {a c}-30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{3} d^{3} x^{3}-674 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}-114 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}+18 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, b^{3} c^{3} x^{3}-236 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-488 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-12 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-272 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{3} c^{2} d x -144 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 a^{2} c \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{4} \sqrt {d b}\, \sqrt {a c}}\)	\(733\)

Input:

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

Output:

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c*(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*d^4*x^4*(d*b)^(1/2)-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*c*d^3*x^4*(d*b)^ 
(1/2)-270*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^2*c^2*d^2*x^4*(d*b)^(1/2)+60*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^3*c^3*d*x^4*(d*b)^(1/2)-9*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^4*c^4*x^4*(d*b)^(1/2)+384*l 
n(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^2*b^2*c*d^3*x^4*(a*c)^(1/2)-30*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c 
)^(1/2)*a^3*d^3*x^3-674*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a^ 
2*b*c*d^2*x^3-114*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a*b^2*c^ 
2*d*x^3+18*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*b^3*c^3*x^3-236 
*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-488*((b*x+a 
)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2-12*((b*x+a)*(d*x+ 
c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a*b^2*c^3*x^2-272*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x-144*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1 
/2)*(a*c)^(1/2)*a^2*b*c^3*x-96*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^( 
1/2)*a^3*c^3)/((b*x+a)*(d*x+c))^(1/2)/x^4/(d*b)^(1/2)/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 4.07 (sec) , antiderivative size = 1529, normalized size of antiderivative = 4.84 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/768*(384*sqrt(b*d)*a^3*b*c^2*d^2*x^4*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*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x 
 + c) + 8*(b^2*c*d + a*b*d^2)*x) - 3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2* 
b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(a*c)*x^4*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)*s 
qrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(48*a^4*c^4 
 - (9*a*b^3*c^4 - 57*a^2*b^2*c^3*d - 337*a^3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 
 + 2*(3*a^2*b^2*c^4 + 122*a^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c 
^4 + 17*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^2*x^4), -1/768*( 
768*sqrt(-b*d)*a^3*b*c^2*d^2*x^4*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b* 
d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2 
)*x)) + 3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^ 
3 - 5*a^4*d^4)*sqrt(a*c)*x^4*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^4*c^4 - (9*a*b^3*c^4 - 57*a^2*b^ 
2*c^3*d - 337*a^3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 + 2*(3*a^2*b^2*c^4 + 122*a 
^3*b*c^3*d + 59*a^4*c^2*d^2)*x^2 + 8*(9*a^3*b*c^4 + 17*a^4*c^3*d)*x)*sqrt( 
b*x + a)*sqrt(d*x + c))/(a^3*c^2*x^4), 1/384*(192*sqrt(b*d)*a^3*b*c^2*d^2* 
x^4*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*x + b*c + 
 a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) ...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^5,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. 3887 vs. \(2 (266) = 532\).

Time = 1.16 (sec) , antiderivative size = 3887, normalized size of antiderivative = 12.30 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/192*(192*sqrt(b*d)*b*d^2*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2 
*c + (b*x + a)*b*d - a*b*d))^2) + 3*(3*sqrt(b*d)*b^5*c^4*abs(b) - 20*sqrt( 
b*d)*a*b^4*c^3*d*abs(b) + 90*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) + 60*sqrt(b* 
d)*a^3*b^2*c*d^3*abs(b) - 5*sqrt(b*d)*a^4*b*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^2*b*c) - 2*(9*sqrt(b*d)*b^19*c^1 
1*abs(b) - 129*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 371*sqrt(b*d)*a^2*b^17*c^9 
*d^2*abs(b) + 581*sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b) - 5494*sqrt(b*d)*a^4*b 
^15*c^7*d^4*abs(b) + 13958*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 19306*sqrt( 
b*d)*a^6*b^13*c^5*d^6*abs(b) + 16154*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 8 
131*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) + 2219*sqrt(b*d)*a^9*b^10*c^2*d^9*ab 
s(b) - 217*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 15*sqrt(b*d)*a^11*b^8*d^11*a 
bs(b) - 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d 
 - a*b*d))^2*b^17*c^10*abs(b) + 702*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s 
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d*abs(b) - 619*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^15 
*c^8*d^2*abs(b) - 5272*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^14*c^7*d^3*abs(b) + 13362*sqrt(b*d)*(sqrt(b 
*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^6*d^ 
4*abs(b) - 7372*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 11.51 (sec) , antiderivative size = 1562, normalized size of antiderivative = 4.94 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^5} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 192*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**4*d - 544*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**5*c**3*d**2*x - 472*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**2*d**3* 
x**2 - 60*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c*d**4*x**3 - 192*sqrt(c + d*x) 
*sqrt(a + b*x)*a**4*b*c**5 - 832*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**4*d 
*x - 1448*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**3*d**2*x**2 - 1408*sqrt(c 
+ d*x)*sqrt(a + b*x)*a**4*b*c**2*d**3*x**3 - 288*sqrt(c + d*x)*sqrt(a + b* 
x)*a**3*b**2*c**5*x - 1000*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**4*d*x* 
*2 - 1576*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**3*d**2*x**3 - 24*sqrt(c 
 + d*x)*sqrt(a + b*x)*a**2*b**3*c**5*x**2 - 192*sqrt(c + d*x)*sqrt(a + b*x 
)*a**2*b**3*c**4*d*x**3 + 36*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**5*x**3 
- 30*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**5*d**5*x**4 + 33 
0*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**4*b*c*d**4*x**4 + 9 
00*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**3*b**2*c**2*d**3*x 
**4 + 420*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*b**3*c**3 
*d**2*x**4 - 102*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqr 
t(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b*...