3.6.75 \(\int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx\) [575]

3.6.75.1 Optimal result

Integrand size = 22, antiderivative size = 436 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=\frac {\left (3 b^2 c^2-6 a b c d-5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{160 a^2 x^4}-\frac {\left (21 b^3 c^3-61 a b^2 c^2 d+51 a^2 b c d^2+5 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c x^3}+\frac {\left (105 b^4 c^4-308 a b^3 c^3 d+262 a^2 b^2 c^2 d^2-20 a^3 b c d^3+25 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{3840 a^4 c^2 x^2}-\frac {\left (315 b^5 c^5-945 a b^4 c^4 d+838 a^2 b^3 c^3 d^2-90 a^3 b^2 c^2 d^3-65 a^4 b c d^4+75 a^5 d^5\right ) \sqrt {a+b x} \sqrt {c+d x}}{7680 a^5 c^3 x}-\frac {(b c+5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{60 a x^5}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}+\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{11/2} c^{7/2}} \]

output

1/512*(-a*d+b*c)^4*(5*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*arctanh(c^(1/2)*(b*x+ 
a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(11/2)/c^(7/2)-1/60*(5*a*d+b*c)*(d*x+c)^ 
(3/2)*(b*x+a)^(1/2)/a/x^5-1/6*(d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6+1/160*(-5*a^ 
2*d^2-6*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/x^4-1/960*(5*a^ 
3*d^3+51*a^2*b*c*d^2-61*a*b^2*c^2*d+21*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2 
)/a^3/c/x^3+1/3840*(25*a^4*d^4-20*a^3*b*c*d^3+262*a^2*b^2*c^2*d^2-308*a*b^ 
3*c^3*d+105*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2/x^2-1/7680*(75*a^ 
5*d^5-65*a^4*b*c*d^4-90*a^3*b^2*c^2*d^3+838*a^2*b^3*c^3*d^2-945*a*b^4*c^4* 
d+315*b^5*c^5)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^3/x
 

3.6.75.2 Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (315 b^5 c^5 x^5-105 a b^4 c^4 x^4 (2 c+9 d x)+2 a^2 b^3 c^3 x^3 \left (84 c^2+308 c d x+419 d^2 x^2\right )-2 a^3 b^2 c^2 x^2 \left (72 c^3+244 c^2 d x+262 c d^2 x^2+45 d^3 x^3\right )+a^4 b c x \left (128 c^4+416 c^3 d x+408 c^2 d^2 x^2+40 c d^3 x^3-65 d^4 x^4\right )+5 a^5 \left (256 c^5+640 c^4 d x+432 c^3 d^2 x^2+8 c^2 d^3 x^3-10 c d^4 x^4+15 d^5 x^5\right )\right )}{7680 a^5 c^3 x^6}+\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{11/2} c^{7/2}} \]

input

Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]
 

output

-1/7680*(Sqrt[a + b*x]*Sqrt[c + d*x]*(315*b^5*c^5*x^5 - 105*a*b^4*c^4*x^4* 
(2*c + 9*d*x) + 2*a^2*b^3*c^3*x^3*(84*c^2 + 308*c*d*x + 419*d^2*x^2) - 2*a 
^3*b^2*c^2*x^2*(72*c^3 + 244*c^2*d*x + 262*c*d^2*x^2 + 45*d^3*x^3) + a^4*b 
*c*x*(128*c^4 + 416*c^3*d*x + 408*c^2*d^2*x^2 + 40*c*d^3*x^3 - 65*d^4*x^4) 
 + 5*a^5*(256*c^5 + 640*c^4*d*x + 432*c^3*d^2*x^2 + 8*c^2*d^3*x^3 - 10*c*d 
^4*x^4 + 15*d^5*x^5)))/(a^5*c^3*x^6) + ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b 
*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] 
)/(512*a^(11/2)*c^(7/2))
 

3.6.75.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {108, 27, 166, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \int \frac {(c+d x)^{3/2} (b c+5 a d+6 b d x)}{x^6 \sqrt {a+b x}}dx-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{12} \left (\frac {\int -\frac {3 \sqrt {c+d x} \left (3 b^2 c^2-6 a b d c-5 a^2 d^2+2 b d (b c-5 a d) x\right )}{2 x^5 \sqrt {a+b x}}dx}{5 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (\frac {\int -\frac {21 b^3 c^3-61 a b^2 d c^2+51 a^2 b d^2 c+5 a^3 d^3+2 b d \left (9 b^2 c^2-26 a b d c+25 a^2 d^2\right ) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {\int \frac {21 b^3 c^3-61 a b^2 d c^2+51 a^2 b d^2 c+5 a^3 d^3+2 b d \left (9 b^2 c^2-26 a b d c+25 a^2 d^2\right ) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {\int \frac {105 b^4 c^4-308 a b^3 d c^3+262 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+25 a^4 d^4+4 b d \left (21 b^3 c^3-61 a b^2 d c^2+51 a^2 b d^2 c+5 a^3 d^3\right ) x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {\int \frac {105 b^4 c^4-308 a b^3 d c^3+262 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+25 a^4 d^4+4 b d \left (21 b^3 c^3-61 a b^2 d c^2+51 a^2 b d^2 c+5 a^3 d^3\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {-\frac {\int \frac {315 b^5 c^5-945 a b^4 d c^4+838 a^2 b^3 d^2 c^3-90 a^3 b^2 d^3 c^2-65 a^4 b d^4 c+75 a^5 d^5+2 b d \left (105 b^4 c^4-308 a b^3 d c^3+262 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+25 a^4 d^4\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {-\frac {\int \frac {315 b^5 c^5-945 a b^4 d c^4+838 a^2 b^3 d^2 c^3-90 a^3 b^2 d^3 c^2-65 a^4 b d^4 c+75 a^5 d^5+2 b d \left (105 b^4 c^4-308 a b^3 d c^3+262 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+25 a^4 d^4\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {-\frac {-\frac {\int \frac {15 (b c-a d)^4 \left (21 b^2 c^2+14 a b d c+5 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {-\frac {-\frac {15 \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {-\frac {-\frac {-\frac {15 \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}}{8 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{12} \left (-\frac {3 \left (-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{4 a x^4}-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{3 a c x^3}-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{2 a c x^2}-\frac {\frac {15 (b c-a d)^4 \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{a c x}}{4 a c}}{6 a c}}{8 a}\right )}{10 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 a d+b c)}{5 a x^5}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{6 x^6}\)

input

Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]
 

output

-1/6*(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^6 + (-1/5*((b*c + 5*a*d)*Sqrt[a + b 
*x]*(c + d*x)^(3/2))/(a*x^5) - (3*(-1/4*((3*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^ 
2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x^4) - (-1/3*((21*b^3*c^3 - 61*a*b^2*c^ 
2*d + 51*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^3) - 
 (-1/2*((105*b^4*c^4 - 308*a*b^3*c^3*d + 262*a^2*b^2*c^2*d^2 - 20*a^3*b*c* 
d^3 + 25*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^2) - (-(((315*b^5*c^ 
5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b* 
c*d^4 + 75*a^5*d^5)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (15*(b*c - a*d 
)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*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*a)))/(10 
*a))/12
 

3.6.75.3.1 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]
 

3.6.75.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(386)=772\).

Time = 0.53 (sec) , antiderivative size = 1068, normalized size of antiderivative = 2.45

method	result	size
default	\(\text {Expression too large to display}\)	\(1068\)

input

int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x,method=_RETURNVERBOSE)
 

output

1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^3*(100*((b*x+a)*(d*x+c))^(1/2)*( 
a*c)^(1/2)*a^5*c*d^4*x^4-256*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^5 
*x-4320*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^3*d^2*x^2+288*((b*x+a)*( 
d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c^5*x^2+420*((b*x+a)*(d*x+c))^(1/2)*(a*c 
)^(1/2)*a*b^4*c^5*x^4-80*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^2*d^3*x 
^3-6400*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^4*d*x-1232*((b*x+a)*(d*x 
+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^4*d*x^4-816*((b*x+a)*(d*x+c))^(1/2)*(a*c) 
^(1/2)*a^4*b*c^3*d^2*x^3+976*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c 
^4*d*x^3-832*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^4*d*x^2-80*((b*x+ 
a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^2*d^3*x^4+1048*((b*x+a)*(d*x+c))^(1/ 
2)*(a*c)^(1/2)*a^3*b^2*c^3*d^2*x^4-90*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^5*b*c*d^5*x^6-75*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^4*x^6-300*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^3*x^6+112 
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^4* 
c^4*d^2*x^6-1050*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*x^6-336*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^5 
*x^3-150*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*d^5*x^5-630*((b*x+a)*(d*x 
+c))^(1/2)*(a*c)^(1/2)*b^5*c^5*x^5-2560*((b*x+a)*(d*x+c))^(1/2)*a^5*c^5*(a 
*c)^(1/2)+130*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c*d^4*x^5+180*(...
 

3.6.75.5 Fricas [A] (verification not implemented)

Time = 6.19 (sec) , antiderivative size = 918, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=\left [\frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, a^{6} d^{6}\right )} \sqrt {a c} x^{6} \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 (1280 \, a^{6} c^{6} + {\left (315 \, a b^{5} c^{6} - 945 \, a^{2} b^{4} c^{5} d + 838 \, a^{3} b^{3} c^{4} d^{2} - 90 \, a^{4} b^{2} c^{3} d^{3} - 65 \, a^{5} b c^{2} d^{4} + 75 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (105 \, a^{2} b^{4} c^{6} - 308 \, a^{3} b^{3} c^{5} d + 262 \, a^{4} b^{2} c^{4} d^{2} - 20 \, a^{5} b c^{3} d^{3} + 25 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (21 \, a^{3} b^{3} c^{6} - 61 \, a^{4} b^{2} c^{5} d + 51 \, a^{5} b c^{4} d^{2} + 5 \, a^{6} c^{3} d^{3}\right )} x^{3} - 16 \, {\left (9 \, a^{4} b^{2} c^{6} - 26 \, a^{5} b c^{5} d - 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, a^{6} c^{4} x^{6}}, -\frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, a^{6} d^{6}\right )} \sqrt {-a c} x^{6} \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 (1280 \, a^{6} c^{6} + {\left (315 \, a b^{5} c^{6} - 945 \, a^{2} b^{4} c^{5} d + 838 \, a^{3} b^{3} c^{4} d^{2} - 90 \, a^{4} b^{2} c^{3} d^{3} - 65 \, a^{5} b c^{2} d^{4} + 75 \, a^{6} c d^{5}\right )} x^{5} - 2 \, {\left (105 \, a^{2} b^{4} c^{6} - 308 \, a^{3} b^{3} c^{5} d + 262 \, a^{4} b^{2} c^{4} d^{2} - 20 \, a^{5} b c^{3} d^{3} + 25 \, a^{6} c^{2} d^{4}\right )} x^{4} + 8 \, {\left (21 \, a^{3} b^{3} c^{6} - 61 \, a^{4} b^{2} c^{5} d + 51 \, a^{5} b c^{4} d^{2} + 5 \, a^{6} c^{3} d^{3}\right )} x^{3} - 16 \, {\left (9 \, a^{4} b^{2} c^{6} - 26 \, a^{5} b c^{5} d - 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \, {\left (a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, a^{6} c^{4} x^{6}}\right ] \]

input

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x, algorithm="fricas")
 

output

[1/30720*(15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^ 
3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + 5*a^6*d^6)*sqrt(a*c)*x^6*l 
og((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*(1280*a^6*c^6 + (315*a*b^5*c^6 - 945*a^2*b^4*c^5*d + 838*a^3*b^3*c^4 
*d^2 - 90*a^4*b^2*c^3*d^3 - 65*a^5*b*c^2*d^4 + 75*a^6*c*d^5)*x^5 - 2*(105* 
a^2*b^4*c^6 - 308*a^3*b^3*c^5*d + 262*a^4*b^2*c^4*d^2 - 20*a^5*b*c^3*d^3 + 
 25*a^6*c^2*d^4)*x^4 + 8*(21*a^3*b^3*c^6 - 61*a^4*b^2*c^5*d + 51*a^5*b*c^4 
*d^2 + 5*a^6*c^3*d^3)*x^3 - 16*(9*a^4*b^2*c^6 - 26*a^5*b*c^5*d - 135*a^6*c 
^4*d^2)*x^2 + 128*(a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c 
))/(a^6*c^4*x^6), -1/15360*(15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c 
^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + 5*a^6*d^ 
6)*sqrt(-a*c)*x^6*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*(12 
80*a^6*c^6 + (315*a*b^5*c^6 - 945*a^2*b^4*c^5*d + 838*a^3*b^3*c^4*d^2 - 90 
*a^4*b^2*c^3*d^3 - 65*a^5*b*c^2*d^4 + 75*a^6*c*d^5)*x^5 - 2*(105*a^2*b^4*c 
^6 - 308*a^3*b^3*c^5*d + 262*a^4*b^2*c^4*d^2 - 20*a^5*b*c^3*d^3 + 25*a^6*c 
^2*d^4)*x^4 + 8*(21*a^3*b^3*c^6 - 61*a^4*b^2*c^5*d + 51*a^5*b*c^4*d^2 + 5* 
a^6*c^3*d^3)*x^3 - 16*(9*a^4*b^2*c^6 - 26*a^5*b*c^5*d - 135*a^6*c^4*d^2)*x 
^2 + 128*(a^5*b*c^6 + 25*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^...
 

3.6.75.6 Sympy [F]

\[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}{x^{7}}\, dx \]

input

integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**7,x)
 

output

Integral(sqrt(a + b*x)*(c + d*x)**(5/2)/x**7, x)
 

3.6.75.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=\text {Exception raised: ValueError} \]

input

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,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
 

3.6.75.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8502 vs. \(2 (386) = 772\).

Time = 3.24 (sec) , antiderivative size = 8502, normalized size of antiderivative = 19.50 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x, algorithm="giac")
 

output

1/7680*(15*(21*sqrt(b*d)*b^7*c^6*abs(b) - 70*sqrt(b*d)*a*b^6*c^5*d*abs(b) 
+ 75*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b) - 20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b 
) - 5*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) - 6*sqrt(b*d)*a^5*b^2*c*d^5*abs(b) 
+ 5*sqrt(b*d)*a^6*b*d^6*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sq 
rt(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^5*b*c^3) - 2*(315*sqrt(b*d)*b^29*c^17*abs(b) - 4725*sqrt 
(b*d)*a*b^28*c^16*d*abs(b) + 32968*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 14 
1816*sqrt(b*d)*a^3*b^26*c^14*d^3*abs(b) + 420148*sqrt(b*d)*a^4*b^25*c^13*d 
^4*abs(b) - 906700*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 1468920*sqrt(b*d)* 
a^6*b^23*c^11*d^6*abs(b) - 1811656*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 17 
01282*sqrt(b*d)*a^8*b^21*c^9*d^8*abs(b) - 1195326*sqrt(b*d)*a^9*b^20*c^8*d 
^9*abs(b) + 595320*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) - 174280*sqrt(b*d)* 
a^11*b^18*c^6*d^11*abs(b) - 4812*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) + 344 
84*sqrt(b*d)*a^13*b^16*c^4*d^13*abs(b) - 18872*sqrt(b*d)*a^14*b^15*c^3*d^1 
4*abs(b) + 5640*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) - 965*sqrt(b*d)*a^16*b 
^13*c*d^16*abs(b) + 75*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 3465*sqrt(b*d)*(s 
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^27*c^16* 
abs(b) + 41160*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^26*c^15*d*abs(b) - 218616*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^25*c^14*d^2*abs(...
 

3.6.75.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^7} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^7} \,d x \]

input

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^7,x)
 

output

int(((a + b*x)^(1/2)*(c + d*x)^(5/2))/x^7, x)