\(\int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\) [400]

Optimal result

Integrand size = 22, antiderivative size = 361 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) \sqrt {a+b x}}{12 a^3 c^3 (b c-a d)^2 (c+d x)^{3/2}}+\frac {d \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^3 c^4 (b c-a d)^3 \sqrt {c+d x}}-\frac {5 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} c^{9/2}} \] Output:

1/4*b*(-7*a^2*d^2+15*b^2*c^2)/a^3/c^2/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^(3/ 
2)-1/2/a/c/x^2/(b*x+a)^(1/2)/(d*x+c)^(3/2)+1/4*(7*a*d+5*b*c)/a^2/c^2/x/(b* 
x+a)^(1/2)/(d*x+c)^(3/2)+1/12*d*(35*a^3*d^3-33*a^2*b*c*d^2-15*a*b^2*c^2*d+ 
45*b^3*c^3)*(b*x+a)^(1/2)/a^3/c^3/(-a*d+b*c)^2/(d*x+c)^(3/2)+1/12*d*(-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)/a^3/c^4/(-a*d+b*c)^3/(d*x+c)^(1/2)-5/4*(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 = 323, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {-45 b^5 c^4 x^2 (c+d x)^2-15 a b^4 c^3 x (c-2 d x) (c+d x)^2+6 a^2 b^3 c^2 (c+d x)^2 \left (c^2+2 c d x+6 d^2 x^2\right )+a^5 d^3 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )-2 a^3 b^2 c d \left (9 c^4-9 c^3 d x-6 c^2 d^2 x^2+111 c d^3 x^3+95 d^4 x^4\right )+a^4 b d^2 \left (18 c^4-48 c^3 d x-237 c^2 d^2 x^2-50 c d^3 x^3+105 d^4 x^4\right )}{12 a^3 c^4 (-b c+a d)^3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {5 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{7/2} c^{9/2}} \] Input:

Integrate[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
 

Output:

(-45*b^5*c^4*x^2*(c + d*x)^2 - 15*a*b^4*c^3*x*(c - 2*d*x)*(c + d*x)^2 + 6* 
a^2*b^3*c^2*(c + d*x)^2*(c^2 + 2*c*d*x + 6*d^2*x^2) + a^5*d^3*(-6*c^3 + 21 
*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3) - 2*a^3*b^2*c*d*(9*c^4 - 9*c^3*d*x 
 - 6*c^2*d^2*x^2 + 111*c*d^3*x^3 + 95*d^4*x^4) + a^4*b*d^2*(18*c^4 - 48*c^ 
3*d*x - 237*c^2*d^2*x^2 - 50*c*d^3*x^3 + 105*d^4*x^4))/(12*a^3*c^4*(-(b*c) 
 + a*d)^3*x^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (5*(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])])/(4*a 
^(7/2)*c^(9/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {114, 27, 168, 27, 169, 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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {\frac {\frac {2 d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}-\frac {2 \int -\frac {15 \left (3 b^2 c^2+6 a b d c+7 a^2 d^2\right ) (b c-a d)^2+2 b d \left (45 b^3 c^3-15 a b^2 d c^2-33 a^2 b d^2 c+35 a^3 d^3\right ) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (15 b^2 c^2-7 a^2 d^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 a d+5 b c}{a c x \sqrt {a+b x} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {15 \left (3 b^2 c^2+6 a b d c+7 a^2 d^2\right ) (b c-a d)^2+2 b d \left (45 b^3 c^3-15 a b^2 d c^2-33 a^2 b d^2 c+35 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (15 b^2 c^2-7 a^2 d^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 a d+5 b c}{a c x \sqrt {a+b x} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {2 d \sqrt {a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {15 (b c-a d)^3 \left (3 b^2 c^2+6 a b d c+7 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (15 b^2 c^2-7 a^2 d^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 a d+5 b c}{a c x \sqrt {a+b x} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {15 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (15 b^2 c^2-7 a^2 d^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 a d+5 b c}{a c x \sqrt {a+b x} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {30 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (15 b^2 c^2-7 a^2 d^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 a d+5 b c}{a c x \sqrt {a+b x} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {\frac {2 b \left (15 b^2 c^2-7 a^2 d^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {\frac {2 d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}+\frac {\frac {2 d \sqrt {a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {30 (b c-a d)^2 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{3 c (b c-a d)}}{a (b c-a d)}}{2 a c}-\frac {7 a d+5 b c}{a c x \sqrt {a+b x} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}\)

Input:

Int[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
 

Output:

-1/2*1/(a*c*x^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (-((5*b*c + 7*a*d)/(a*c*x 
*Sqrt[a + b*x]*(c + d*x)^(3/2))) - ((2*b*(15*b^2*c^2 - 7*a^2*d^2))/(a*(b*c 
 - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + ((2*d*(45*b^3*c^3 - 15*a*b^2*c^2* 
d - 33*a^2*b*c*d^2 + 35*a^3*d^3)*Sqrt[a + b*x])/(3*c*(b*c - a*d)*(c + d*x) 
^(3/2)) + ((2*d*(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])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (3 
0*(b*c - a*d)^2*(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])])/(Sqrt[a]*c^(3/2)))/(3*c*(b*c - a*d)))/ 
(a*(b*c - a*d)))/(2*a*c))/(4*a*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 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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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. \(2215\) vs. \(2(317)=634\).

Time = 0.34 (sec) , antiderivative size = 2216, normalized size of antiderivative = 6.14

method	result	size
default	\(\text {Expression too large to display}\)	\(2216\)

Input:

int(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-36*a^4*b*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+36*a^3*b^2*c^ 
5*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-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^2*c*d^6*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^3*c^2*d^5*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^4*c^3*d^4* 
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 
^5*c^4*d^3*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*b*c*d^6*x^4-360*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^5*x^4+210*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^4*x^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^2*b^4*c^4*d^3*x^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^6*d^7*x^4-45 
*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^6*c^7*x 
^3+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^2* 
b^4*c^6*d*x^2+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^5*c^5*d^2*x^4-345*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^2*d^5*x^3-45*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^4*b^2*c^3*d^4*x^3+150*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^4*d^3*x^3+120*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^5*d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 991 vs. \(2 (317) = 634\).

Time = 4.92 (sec) , antiderivative size = 2002, normalized size of antiderivative = 5.55 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/48*(15*((3*b^6*c^5*d^2 - 3*a*b^5*c^4*d^3 - 2*a^2*b^4*c^3*d^4 - 6*a^3*b^ 
3*c^2*d^5 + 15*a^4*b^2*c*d^6 - 7*a^5*b*d^7)*x^5 + (6*b^6*c^6*d - 3*a*b^5*c 
^5*d^2 - 7*a^2*b^4*c^4*d^3 - 14*a^3*b^3*c^3*d^4 + 24*a^4*b^2*c^2*d^5 + a^5 
*b*c*d^6 - 7*a^6*d^7)*x^4 + (3*b^6*c^7 + 3*a*b^5*c^6*d - 8*a^2*b^4*c^5*d^2 
 - 10*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 + 23*a^5*b*c^2*d^5 - 14*a^6*c*d^ 
6)*x^3 + (3*a*b^5*c^7 - 3*a^2*b^4*c^6*d - 2*a^3*b^3*c^5*d^2 - 6*a^4*b^2*c^ 
4*d^3 + 15*a^5*b*c^3*d^4 - 7*a^6*c^2*d^5)*x^2)*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*(6*a^3*b^3 
*c^7 - 18*a^4*b^2*c^6*d + 18*a^5*b*c^5*d^2 - 6*a^6*c^4*d^3 - (45*a*b^5*c^5 
*d^2 - 30*a^2*b^4*c^4*d^3 - 36*a^3*b^3*c^3*d^4 + 190*a^4*b^2*c^2*d^5 - 105 
*a^5*b*c*d^6)*x^4 - (90*a*b^5*c^6*d - 45*a^2*b^4*c^5*d^2 - 84*a^3*b^3*c^4* 
d^3 + 222*a^4*b^2*c^3*d^4 + 50*a^5*b*c^2*d^5 - 105*a^6*c*d^6)*x^3 - (45*a* 
b^5*c^7 - 66*a^3*b^3*c^5*d^2 - 12*a^4*b^2*c^4*d^3 + 237*a^5*b*c^3*d^4 - 14 
0*a^6*c^2*d^5)*x^2 - 3*(5*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d - 6*a^4*b^2*c^5*d^ 
2 + 16*a^5*b*c^4*d^3 - 7*a^6*c^3*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^ 
4*b^4*c^8*d^2 - 3*a^5*b^3*c^7*d^3 + 3*a^6*b^2*c^6*d^4 - a^7*b*c^5*d^5)*x^5 
 + (2*a^4*b^4*c^9*d - 5*a^5*b^3*c^8*d^2 + 3*a^6*b^2*c^7*d^3 + a^7*b*c^6*d^ 
4 - a^8*c^5*d^5)*x^4 + (a^4*b^4*c^10 - a^5*b^3*c^9*d - 3*a^6*b^2*c^8*d^2 + 
 5*a^7*b*c^7*d^3 - 2*a^8*c^6*d^4)*x^3 + (a^5*b^3*c^10 - 3*a^6*b^2*c^9*d...
 

Sympy [F]

\[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/x**3/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
 

Output:

Integral(1/(x**3*(a + b*x)**(3/2)*(c + d*x)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1482 vs. \(2 (317) = 634\).

Time = 3.30 (sec) , antiderivative size = 1482, normalized size of antiderivative = 4.11 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

4*sqrt(b*d)*b^6/((a^3*b^2*c^2*abs(b) - 2*a^4*b*c*d*abs(b) + a^5*d^2*abs(b) 
)*(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)) + 2/3*sqrt(b*x + a)*((14*b^6*c^7*d^6*abs(b) - 37*a*b^5*c^6*d^ 
7*abs(b) + 32*a^2*b^4*c^5*d^8*abs(b) - 9*a^3*b^3*c^4*d^9*abs(b))*(b*x + a) 
/(b^7*c^13*d - 5*a*b^6*c^12*d^2 + 10*a^2*b^5*c^11*d^3 - 10*a^3*b^4*c^10*d^ 
4 + 5*a^4*b^3*c^9*d^5 - a^5*b^2*c^8*d^6) + 3*(5*b^7*c^8*d^5*abs(b) - 18*a* 
b^6*c^7*d^6*abs(b) + 24*a^2*b^5*c^6*d^7*abs(b) - 14*a^3*b^4*c^5*d^8*abs(b) 
 + 3*a^4*b^3*c^4*d^9*abs(b))/(b^7*c^13*d - 5*a*b^6*c^12*d^2 + 10*a^2*b^5*c 
^11*d^3 - 10*a^3*b^4*c^10*d^4 + 5*a^4*b^3*c^9*d^5 - a^5*b^2*c^8*d^6))/(b^2 
*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5/4*(3*sqrt(b*d)*b^4*c^2 + 6*sqrt(b*d) 
*a*b^3*c*d + 7*sqrt(b*d)*a^2*b^2*d^2)*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*c^4*abs(b)) + 1/2*(7*sqrt(b*d)*b^10*c^5 - 17*s 
qrt(b*d)*a*b^9*c^4*d - 2*sqrt(b*d)*a^2*b^8*c^3*d^2 + 38*sqrt(b*d)*a^3*b^7* 
c^2*d^3 - 37*sqrt(b*d)*a^4*b^6*c*d^4 + 11*sqrt(b*d)*a^5*b^5*d^5 - 21*sqrt( 
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^8 
*c^4 - 16*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^7*c^3*d + 62*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^6*c^2*d^2 + 8*sqrt(b*d)*(sqrt(b*d)*sq 
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*c*d^3 - 33...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^3\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x)
 

Output:

int(1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 3.45 (sec) , antiderivative size = 5118, normalized size of antiderivative = 14.18 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)
 

Output:

(315*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*s 
qrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**6* 
c**2*d**6*x**2 + 630*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*s 
qrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt 
(c + d*x))*a**6*c*d**7*x**3 + 315*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqr 
t(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**6*d**8*x**4 - 570*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*b*c**3*d**5*x**2 - 1140*sqrt(c)*sq 
rt(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*b*c**2*d**6*x** 
3 - 570*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*b*c*d**7*x**4 + 45*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)*sqr 
t(c + d*x))*a**4*b**2*c**4*d**4*x**2 + 90*sqrt(c)*sqrt(a)*sqrt(a + b*x)*lo 
g( - 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**2*c**3*d**5*x**3 + 45*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**4*b**2*c**2*d**6...