\(\int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [420]

Optimal result

Integrand size = 22, antiderivative size = 462 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {b \left (35 b^2 c^2-6 a b c d-21 a^2 d^2\right )}{12 a^3 c^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {7 (b c+a d)}{4 a^2 c^2 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {b \left (35 b^3 c^3-55 a b^2 c^2 d-3 a^2 b c d^2+7 a^3 d^3\right )}{4 a^4 c^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (105 b^4 c^4-200 a b^3 c^3 d+18 a^2 b^2 c^2 d^2+48 a^3 b c d^3-35 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {d (b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^4 c^4 (b c-a d)^4 \sqrt {c+d x}}-\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2} c^{9/2}} \] Output:

1/12*b*(-21*a^2*d^2-6*a*b*c*d+35*b^2*c^2)/a^3/c^2/(-a*d+b*c)/(b*x+a)^(3/2) 
/(d*x+c)^(3/2)-1/2/a/c/x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2)+7/4*(a*d+b*c)/a^2/c 
^2/x/(b*x+a)^(3/2)/(d*x+c)^(3/2)+1/4*b*(7*a^3*d^3-3*a^2*b*c*d^2-55*a*b^2*c 
^2*d+35*b^3*c^3)/a^4/c^2/(-a*d+b*c)^2/(b*x+a)^(1/2)/(d*x+c)^(3/2)+1/12*d*( 
-35*a^4*d^4+48*a^3*b*c*d^3+18*a^2*b^2*c^2*d^2-200*a*b^3*c^3*d+105*b^4*c^4) 
*(b*x+a)^(1/2)/a^4/c^3/(-a*d+b*c)^3/(d*x+c)^(3/2)+1/12*d*(a*d+b*c)*(105*a^ 
4*d^4-340*a^3*b*c*d^3+406*a^2*b^2*c^2*d^2-340*a*b^3*c^3*d+105*b^4*c^4)*(b* 
x+a)^(1/2)/a^4/c^4/(-a*d+b*c)^4/(d*x+c)^(1/2)-5/4*(7*a^2*d^2+10*a*b*c*d+7* 
b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(9 
/2)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {105 b^7 c^5 x^3 (c+d x)^2+5 a b^6 c^4 x^2 (28 c-47 d x) (c+d x)^2+3 a^2 b^5 c^3 x (c+d x)^2 \left (7 c^2-106 c d x+22 d^2 x^2\right )-3 a^3 b^4 c^2 (c+d x)^2 \left (2 c^3+17 c^2 d x-32 c d^2 x^2-22 d^3 x^3\right )+a^7 d^4 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )+3 a^6 b d^3 \left (8 c^4-21 c^3 d x-92 c^2 d^2 x^2+15 c d^3 x^3+70 d^4 x^4\right )+a^4 b^3 c d \left (24 c^5+42 c^4 d x+168 c^3 d^2 x^2+207 c^2 d^3 x^3-186 c d^4 x^4-235 d^5 x^5\right )-3 a^5 b^2 d^2 \left (12 c^5-14 c^4 d x+4 c^3 d^2 x^2+183 c^2 d^3 x^3+110 c d^4 x^4-35 d^5 x^5\right )}{12 a^4 c^4 (b c-a d)^4 x^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{9/2} c^{9/2}} \] Input:

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

Output:

(105*b^7*c^5*x^3*(c + d*x)^2 + 5*a*b^6*c^4*x^2*(28*c - 47*d*x)*(c + d*x)^2 
 + 3*a^2*b^5*c^3*x*(c + d*x)^2*(7*c^2 - 106*c*d*x + 22*d^2*x^2) - 3*a^3*b^ 
4*c^2*(c + d*x)^2*(2*c^3 + 17*c^2*d*x - 32*c*d^2*x^2 - 22*d^3*x^3) + a^7*d 
^4*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3) + 3*a^6*b*d^3*(8*c^ 
4 - 21*c^3*d*x - 92*c^2*d^2*x^2 + 15*c*d^3*x^3 + 70*d^4*x^4) + a^4*b^3*c*d 
*(24*c^5 + 42*c^4*d*x + 168*c^3*d^2*x^2 + 207*c^2*d^3*x^3 - 186*c*d^4*x^4 
- 235*d^5*x^5) - 3*a^5*b^2*d^2*(12*c^5 - 14*c^4*d*x + 4*c^3*d^2*x^2 + 183* 
c^2*d^3*x^3 + 110*c*d^4*x^4 - 35*d^5*x^5))/(12*a^4*c^4*(b*c - a*d)^4*x^2*( 
a + b*x)^(3/2)*(c + d*x)^(3/2)) - (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)* 
ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(4*a^(9/2)*c^(9/ 
2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {114, 27, 168, 27, 169, 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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {5 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right )+56 b d (b c+a d) x}{2 x (a+b x)^{5/2} (c+d x)^{5/2}}dx}{a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {5 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right )+56 b d (b c+a d) x}{x (a+b x)^{5/2} (c+d x)^{5/2}}dx}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {\frac {2 \int \frac {3 \left (5 (b c-a d) \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right )+2 b d \left (35 b^2 c^2-6 a b d c-21 a^2 d^2\right ) x\right )}{2 x (a+b x)^{3/2} (c+d x)^{5/2}}dx}{3 a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {5 (b c-a d) \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right )+2 b d \left (35 b^2 c^2-6 a b d c-21 a^2 d^2\right ) x}{x (a+b x)^{3/2} (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {\frac {\frac {2 \int \frac {5 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right ) (b c-a d)^2+4 b d \left (35 b^3 c^3-55 a b^2 d c^2-3 a^2 b d^2 c+7 a^3 d^3\right ) x}{2 x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {5 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right ) (b c-a d)^2+4 b d \left (35 b^3 c^3-55 a b^2 d c^2-3 a^2 b d^2 c+7 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {2 d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{3 c (c+d x)^{3/2} (b c-a d)}-\frac {2 \int -\frac {15 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right ) (b c-a d)^3+2 b d \left (105 b^4 c^4-200 a b^3 d c^3+18 a^2 b^2 d^2 c^2+48 a^3 b d^3 c-35 a^4 d^4\right ) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\int \frac {15 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right ) (b c-a d)^3+2 b d \left (105 b^4 c^4-200 a b^3 d c^3+18 a^2 b^2 d^2 c^2+48 a^3 b d^3 c-35 a^4 d^4\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\frac {2 d \sqrt {a+b x} (a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {15 (b c-a d)^4 \left (7 b^2 c^2+10 a b d c+7 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\frac {15 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} (a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {\frac {30 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} (a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {\frac {2 b \left (-21 a^2 d^2-6 a b c d+35 b^2 c^2\right )}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac {\frac {2 b \left (7 a^3 d^3-3 a^2 b c d^2-55 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {\frac {\frac {2 d \sqrt {a+b x} (a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {30 (b c-a d)^3 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-35 a^4 d^4+48 a^3 b c d^3+18 a^2 b^2 c^2 d^2-200 a b^3 c^3 d+105 b^4 c^4\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}}{a (b c-a d)}}{2 a c}-\frac {7 (a d+b c)}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}}{4 a c}-\frac {1}{2 a c x^2 (a+b x)^{3/2} (c+d x)^{3/2}}\)

Input:

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

Output:

-1/2*1/(a*c*x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - ((-7*(b*c + a*d))/(a*c* 
x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - ((2*b*(35*b^2*c^2 - 6*a*b*c*d - 21*a^ 
2*d^2))/(3*a*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + ((2*b*(35*b^3* 
c^3 - 55*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 7*a^3*d^3))/(a*(b*c - a*d)*Sqrt[a + 
 b*x]*(c + d*x)^(3/2)) + ((2*d*(105*b^4*c^4 - 200*a*b^3*c^3*d + 18*a^2*b^2 
*c^2*d^2 + 48*a^3*b*c*d^3 - 35*a^4*d^4)*Sqrt[a + b*x])/(3*c*(b*c - a*d)*(c 
 + d*x)^(3/2)) + ((2*d*(b*c + a*d)*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^ 
2*b^2*c^2*d^2 - 340*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[a + b*x])/(c*(b*c - a* 
d)*Sqrt[c + d*x]) - (30*(b*c - a*d)^3*(7*b^2*c^2 + 10*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)))/(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. \(3391\) vs. \(2(412)=824\).

Time = 0.34 (sec) , antiderivative size = 3392, normalized size of antiderivative = 7.34

method	result	size
default	\(\text {Expression too large to display}\)	\(3392\)

Input:

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

Output:

-1/24/c^4/a^4*(330*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^3*c^3*d^5*x^4+510*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^4*c^4*d^4*x^4+330*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^5*c^5*d^3*x^4-840*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^6*c^6*d^2*x^4+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*b^7*c^7 
*d*x^4-330*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^7*b*c^2*d^6*x^3-270*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^6*b^2*c^3*d^5*x^3+390*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^3*c^4*d^4*x^3+390*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^4*c^5*d^3*x^3-270*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^3*b^5*c^6*d^2*x^3 
-330*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 
^6*c^7*d*x^3-270*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^7*b*c^3*d^5*x^2+135*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*b^2*c^4*d^4*x^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^5*b^3*c^5*d^3*x^2+135*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^4*c^6*d^2*x^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^3*b^5*c^7*d* 
x^2-210*a^5*b^2*d^7*x^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*b^7*c^5...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (412) = 824\).

Time = 11.11 (sec) , antiderivative size = 2996, normalized size of antiderivative = 6.48 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(15*((7*b^8*c^6*d^2 - 18*a*b^7*c^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b 
^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^6 + 2 
*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*c^4*d^4 
+ 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^8) 
*x^5 + (7*b^8*c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d 
^3 + 34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*a^7 
*b*c*d^7 + 7*a^8*d^8)*x^4 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5* 
c^6*d^2 + 13*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11 
*a^7*b*c^2*d^6 + 7*a^8*c*d^7)*x^3 + (7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9* 
a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*d^3 + 9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 
 + 7*a^8*c^2*d^6)*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^4*b^4*c^8 - 24*a^5*b^3*c^7*d 
+ 36*a^6*b^2*c^6*d^2 - 24*a^7*b*c^5*d^3 + 6*a^8*c^4*d^4 - (105*a*b^7*c^6*d 
^2 - 235*a^2*b^6*c^5*d^3 + 66*a^3*b^5*c^4*d^4 + 66*a^4*b^4*c^3*d^5 - 235*a 
^5*b^3*c^2*d^6 + 105*a^6*b^2*c*d^7)*x^5 - 6*(35*a*b^7*c^7*d - 55*a^2*b^6*c 
^6*d^2 - 31*a^3*b^5*c^5*d^3 + 38*a^4*b^4*c^4*d^4 - 31*a^5*b^3*c^3*d^5 - 55 
*a^6*b^2*c^2*d^6 + 35*a^7*b*c*d^7)*x^4 - 3*(35*a*b^7*c^8 + 15*a^2*b^6*c^7* 
d - 183*a^3*b^5*c^6*d^2 + 69*a^4*b^4*c^5*d^3 + 69*a^5*b^3*c^4*d^4 - 183*a^ 
6*b^2*c^3*d^5 + 15*a^7*b*c^2*d^6 + 35*a^8*c*d^7)*x^3 - 4*(35*a^2*b^6*c^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),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. 1901 vs. \(2 (412) = 824\).

Time = 9.94 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.11 \[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*sqrt(b*x + a)*((17*b^7*c^8*d^7*abs(b) - 60*a*b^6*c^7*d^8*abs(b) + 78* 
a^2*b^5*c^6*d^9*abs(b) - 44*a^3*b^4*c^5*d^10*abs(b) + 9*a^4*b^3*c^4*d^11*a 
bs(b))*(b*x + a)/(b^9*c^15*d - 7*a*b^8*c^14*d^2 + 21*a^2*b^7*c^13*d^3 - 35 
*a^3*b^6*c^12*d^4 + 35*a^4*b^5*c^11*d^5 - 21*a^5*b^4*c^10*d^6 + 7*a^6*b^3* 
c^9*d^7 - a^7*b^2*c^8*d^8) + 9*(2*b^8*c^9*d^6*abs(b) - 9*a*b^7*c^8*d^7*abs 
(b) + 16*a^2*b^6*c^7*d^8*abs(b) - 14*a^3*b^5*c^6*d^9*abs(b) + 6*a^4*b^4*c^ 
5*d^10*abs(b) - a^5*b^3*c^4*d^11*abs(b))/(b^9*c^15*d - 7*a*b^8*c^14*d^2 + 
21*a^2*b^7*c^13*d^3 - 35*a^3*b^6*c^12*d^4 + 35*a^4*b^5*c^11*d^5 - 21*a^5*b 
^4*c^10*d^6 + 7*a^6*b^3*c^9*d^7 - a^7*b^2*c^8*d^8))/(b^2*c + (b*x + a)*b*d 
 - a*b*d)^(3/2) + 4/3*(9*sqrt(b*d)*b^11*c^3 - 35*sqrt(b*d)*a*b^10*c^2*d + 
43*sqrt(b*d)*a^2*b^9*c*d^2 - 17*sqrt(b*d)*a^3*b^8*d^3 - 18*sqrt(b*d)*(sqrt 
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^9*c^2 + 54* 
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^8*c*d - 36*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^7*d^2 + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq 
rt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^7*c - 15*sqrt(b*d)*(sqrt(b*d)*sqrt( 
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^6*d)/((a^4*b^3*c^3*a 
bs(b) - 3*a^5*b^2*c^2*d*abs(b) + 3*a^6*b*c*d^2*abs(b) - a^7*d^3*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)^3) - 5/4*(7*sqrt(b*d)*b^4*c^2 + 10*sqrt(b*d)*a*b^3*c*d + 7*sqrt...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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