\(\int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx\) [222]

Optimal result

Integrand size = 22, antiderivative size = 340 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{9/2}} \] Output:

-1/40*(a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^4-1/240*(3*b^2*c/a+12*b* 
d-7*a*d^2/c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^3+1/960*(-35*a^3*d^3+61*a^2*b 
*c*d^2-9*a*b^2*c^2*d+15*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^3/x^2-1 
/1920*(-105*a^4*d^4+190*a^3*b*c*d^3-36*a^2*b^2*c^2*d^2-30*a*b^3*c^3*d+45*b 
^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4/x-1/5*(b*x+a)^(3/2)*(d*x+c)^(1 
/2)/x^5+1/128*(-a*d+b*c)^3*(7*a^2*d^2+6*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2) 
*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (45 b^4 c^4 x^4-30 a b^3 c^3 x^3 (c+d x)+6 a^2 b^2 c^2 x^2 \left (4 c^2+3 c d x-6 d^2 x^2\right )+2 a^3 b c x \left (264 c^3+48 c^2 d x-61 c d^2 x^2+95 d^3 x^3\right )+a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )\right )}{1920 a^3 c^4 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{7/2} c^{9/2}} \] Input:

Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]
 

Output:

-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(45*b^4*c^4*x^4 - 30*a*b^3*c^3*x^3*(c 
 + d*x) + 6*a^2*b^2*c^2*x^2*(4*c^2 + 3*c*d*x - 6*d^2*x^2) + 2*a^3*b*c*x*(2 
64*c^3 + 48*c^2*d*x - 61*c*d^2*x^2 + 95*d^3*x^3) + a^4*(384*c^4 + 48*c^3*d 
*x - 56*c^2*d^2*x^2 + 70*c*d^3*x^3 - 105*d^4*x^4)))/(a^3*c^4*x^5) + ((b*c 
- a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x 
])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(7/2)*c^(9/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.08, 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, 168, 27, 168, 27, 168, 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.

\(\displaystyle \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx\)

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \int \frac {\sqrt {a+b x} (3 b c+a d+4 b d x)}{x^5 \sqrt {c+d x}}dx-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{10} \left (\frac {\int \frac {3 b^2 c^2+12 a b d c-7 a^2 d^2+2 b d (7 b c-3 a d) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {\int \frac {3 b^2 c^2+12 a b d c-7 a^2 d^2+2 b d (7 b c-3 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3+4 b d \left (3 b^2 c^2+12 a b d c-7 a^2 d^2\right ) x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3+4 b d \left (3 b^2 c^2+12 a b d c-7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {\int \frac {45 b^4 c^4-30 a b^3 d c^3-36 a^2 b^2 d^2 c^2+190 a^3 b d^3 c-105 a^4 d^4+2 b d \left (15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {\int \frac {45 b^4 c^4-30 a b^3 d c^3-36 a^2 b^2 d^2 c^2+190 a^3 b d^3 c-105 a^4 d^4+2 b d \left (15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {-\frac {\int \frac {15 (b c-a d)^3 \left (3 b^2 c^2+6 a b d c+7 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {-\frac {15 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {-\frac {15 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}-\frac {\frac {15 (b c-a d)^3 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\)

Input:

Int[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]
 

Output:

-1/5*((a + b*x)^(3/2)*Sqrt[c + d*x])/x^5 + (-1/4*((3*b*c + a*d)*Sqrt[a + b 
*x]*Sqrt[c + d*x])/(c*x^4) + (-1/3*(((3*b^2*c)/a + 12*b*d - (7*a*d^2)/c)*S 
qrt[a + b*x]*Sqrt[c + d*x])/x^3 - (-1/2*((15*b^3*c^3 - 9*a*b^2*c^2*d + 61* 
a^2*b*c*d^2 - 35*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^2) - (-(((45 
*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4 
*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (15*(b*c - a*d)^3*(3*b^2*c^2 
 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c 
+ d*x])])/(a^(3/2)*c^(3/2)))/(4*a*c))/(6*a*c))/(8*c))/10
 

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 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 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 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. \(812\) vs. \(2(296)=592\).

Time = 0.23 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.39

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (105 \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^{5} x^{5}-225 \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^{4} x^{5}+90 \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^{3} x^{5}+30 \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^{2} x^{5}+45 \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} d \,x^{5}-45 \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^{5} x^{5}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{4} d^{4} x^{4}+380 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} b c \,d^{3} x^{4}-72 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}-60 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a \,b^{3} c^{3} d \,x^{4}+90 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{4} c^{4} x^{4}+140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}-244 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+36 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}-60 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+192 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+48 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x +1056 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{3} c^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{5} \sqrt {a c}}\)	\(813\)

Input:

int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(105*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^5*x^5-225*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^4*x^5+90*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^3*x^5+ 
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^2*b^3 
*c^3*d^2*x^5+45*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*d*x^5-45*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^5*x^5-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*d^4 
*x^4+380*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c*d^3*x^4-72*(a*c)^(1/2 
)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^2*c^2*d^2*x^4-60*(a*c)^(1/2)*((b*x+a)*(d*x 
+c))^(1/2)*a*b^3*c^3*d*x^4+90*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^4*c^4* 
x^4+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3-244*((b*x+a)*(d* 
x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3+36*((b*x+a)*(d*x+c))^(1/2)*(a*c) 
^(1/2)*a^2*b^2*c^3*d*x^3-60*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^3*c^4* 
x^3-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2+192*((b*x+a)*( 
d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2+48*((b*x+a)*(d*x+c))^(1/2)*(a*c) 
^(1/2)*a^2*b^2*c^4*x^2+96*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^3*d*x+ 
1056*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^4*x+768*((b*x+a)*(d*x+c)) 
^(1/2)*a^4*c^4*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^5/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 3.06 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\left [-\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (384 \, a^{5} c^{5} + {\left (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{4} c^{5} x^{5}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (384 \, a^{5} c^{5} + {\left (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{4} c^{5} x^{5}}\right ] \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x, algorithm="fricas")
 

Output:

[-1/7680*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^ 
2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*sqrt(a*c)*x^5*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*(384*a^5*c^5 + (4 
5*a*b^4*c^5 - 30*a^2*b^3*c^4*d - 36*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 
105*a^5*c*d^4)*x^4 - 2*(15*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 61*a^4*b*c^3*d^ 
2 - 35*a^5*c^2*d^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^ 
2)*x^2 + 48*(11*a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^ 
4*c^5*x^5), -1/3840*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6 
*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*sqrt(-a*c)*x^5*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*(384*a^5*c^5 + (45*a*b^4*c^5 - 30 
*a^2*b^3*c^4*d - 36*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x 
^4 - 2*(15*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 61*a^4*b*c^3*d^2 - 35*a^5*c^2*d 
^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2 + 48*(11* 
a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^5)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}{x^{6}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(d*x+c)**(1/2)/x**6,x)
 

Output:

Integral((a + b*x)**(3/2)*sqrt(c + d*x)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,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. 5867 vs. \(2 (296) = 592\).

Time = 9.48 (sec) , antiderivative size = 5867, normalized size of antiderivative = 17.26 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x, algorithm="giac")
 

Output:

1/1920*b^5*(15*(3*sqrt(b*d)*b^5*c^5*abs(b) - 3*sqrt(b*d)*a*b^4*c^4*d*abs(b 
) - 2*sqrt(b*d)*a^2*b^3*c^3*d^2*abs(b) - 6*sqrt(b*d)*a^3*b^2*c^2*d^3*abs(b 
) + 15*sqrt(b*d)*a^4*b*c*d^4*abs(b) - 7*sqrt(b*d)*a^5*d^5*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^3*b^6*c^4) - 2*(45*sqr 
t(b*d)*b^23*c^14*abs(b) - 480*sqrt(b*d)*a*b^22*c^13*d*abs(b) + 2289*sqrt(b 
*d)*a^2*b^21*c^12*d^2*abs(b) - 6200*sqrt(b*d)*a^3*b^20*c^11*d^3*abs(b) + 9 
425*sqrt(b*d)*a^4*b^19*c^10*d^4*abs(b) - 3720*sqrt(b*d)*a^5*b^18*c^9*d^5*a 
bs(b) - 18075*sqrt(b*d)*a^6*b^17*c^8*d^6*abs(b) + 49872*sqrt(b*d)*a^7*b^16 
*c^7*d^7*abs(b) - 71865*sqrt(b*d)*a^8*b^15*c^6*d^8*abs(b) + 68880*sqrt(b*d 
)*a^9*b^14*c^5*d^9*abs(b) - 46125*sqrt(b*d)*a^10*b^13*c^4*d^10*abs(b) + 21 
480*sqrt(b*d)*a^11*b^12*c^3*d^11*abs(b) - 6661*sqrt(b*d)*a^12*b^11*c^2*d^1 
2*abs(b) + 1240*sqrt(b*d)*a^13*b^10*c*d^13*abs(b) - 105*sqrt(b*d)*a^14*b^9 
*d^14*abs(b) - 405*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x 
+ a)*b*d - a*b*d))^2*b^21*c^13*abs(b) + 3045*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^20*c^12*d*abs(b) - 9150 
*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^19*c^11*d^2*abs(b) + 10550*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^3*b^18*c^10*d^3*abs(b) + 12625*sqr 
t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x^6} \,d x \] Input:

int(((a + b*x)^(3/2)*(c + d*x)^(1/2))/x^6,x)
 

Output:

int(((a + b*x)^(3/2)*(c + d*x)^(1/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 33.64 (sec) , antiderivative size = 1655, normalized size of antiderivative = 4.87 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x)
 

Output:

( - 768*sqrt(c + d*x)*sqrt(a + b*x)*a**6*c**5*d - 96*sqrt(c + d*x)*sqrt(a 
+ b*x)*a**6*c**4*d**2*x + 112*sqrt(c + d*x)*sqrt(a + b*x)*a**6*c**3*d**3*x 
**2 - 140*sqrt(c + d*x)*sqrt(a + b*x)*a**6*c**2*d**4*x**3 + 210*sqrt(c + d 
*x)*sqrt(a + b*x)*a**6*c*d**5*x**4 - 768*sqrt(c + d*x)*sqrt(a + b*x)*a**5* 
b*c**6 - 1152*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b*c**5*d*x - 80*sqrt(c + d* 
x)*sqrt(a + b*x)*a**5*b*c**4*d**2*x**2 + 104*sqrt(c + d*x)*sqrt(a + b*x)*a 
**5*b*c**3*d**3*x**3 - 170*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b*c**2*d**4*x* 
*4 - 1056*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**2*c**6*x - 240*sqrt(c + d*x) 
*sqrt(a + b*x)*a**4*b**2*c**5*d*x**2 + 208*sqrt(c + d*x)*sqrt(a + b*x)*a** 
4*b**2*c**4*d**2*x**3 - 308*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**2*c**3*d** 
3*x**4 - 48*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**3*c**6*x**2 + 24*sqrt(c + 
d*x)*sqrt(a + b*x)*a**3*b**3*c**5*d*x**3 + 132*sqrt(c + d*x)*sqrt(a + b*x) 
*a**3*b**3*c**4*d**2*x**4 + 60*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c**6* 
x**3 - 30*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c**5*d*x**4 - 90*sqrt(c + 
d*x)*sqrt(a + b*x)*a*b**5*c**6*x**4 + 105*sqrt(c)*sqrt(a)*log( - sqrt(2*sq 
rt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt( 
b)*sqrt(c + d*x))*a**6*d**6*x**5 - 120*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*b*c*d**5*x**5 - 135*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...