\(\int \frac {x (a x+b x^2)^{5/2}}{c+d x} \, dx\) [115]

Optimal result

Integrand size = 22, antiderivative size = 493 \[ \int \frac {x \left (a x+b x^2\right )^{5/2}}{c+d x} \, dx=-\frac {\left (512 b^5 c^5-1152 a b^4 c^4 d+704 a^2 b^3 c^3 d^2-40 a^3 b^2 c^2 d^3-12 a^4 b c d^4-5 a^5 d^5\right ) \sqrt {a x+b x^2}}{512 b^3 d^6}+\frac {\left (384 b^4 c^4-832 a b^3 c^3 d+472 a^2 b^2 c^2 d^2-12 a^3 b c d^3-5 a^4 d^4\right ) x \sqrt {a x+b x^2}}{768 b^2 d^5}-\frac {\left (320 b^3 c^3-680 a b^2 c^2 d+372 a^2 b c d^2-5 a^3 d^3\right ) x^2 \sqrt {a x+b x^2}}{960 b d^4}+\frac {\left (40 b^2 c^2-52 a b c d+5 a^2 d^2\right ) x^3 \sqrt {a x+b x^2}}{160 d^3}-\frac {(12 b c-5 a d) x^2 \left (a x+b x^2\right )^{3/2}}{60 d^2}+\frac {x \left (a x+b x^2\right )^{5/2}}{6 d}+\frac {\left (1024 b^6 c^6-2560 a b^5 c^5 d+1920 a^2 b^4 c^4 d^2-320 a^3 b^3 c^3 d^3-40 a^4 b^2 c^2 d^4-12 a^5 b c d^5-5 a^6 d^6\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{7/2} d^7}-\frac {2 c^{7/2} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^7} \] Output:

-1/512*(-5*a^5*d^5-12*a^4*b*c*d^4-40*a^3*b^2*c^2*d^3+704*a^2*b^3*c^3*d^2-1 
152*a*b^4*c^4*d+512*b^5*c^5)*(b*x^2+a*x)^(1/2)/b^3/d^6+1/768*(-5*a^4*d^4-1 
2*a^3*b*c*d^3+472*a^2*b^2*c^2*d^2-832*a*b^3*c^3*d+384*b^4*c^4)*x*(b*x^2+a* 
x)^(1/2)/b^2/d^5-1/960*(-5*a^3*d^3+372*a^2*b*c*d^2-680*a*b^2*c^2*d+320*b^3 
*c^3)*x^2*(b*x^2+a*x)^(1/2)/b/d^4+1/160*(5*a^2*d^2-52*a*b*c*d+40*b^2*c^2)* 
x^3*(b*x^2+a*x)^(1/2)/d^3-1/60*(-5*a*d+12*b*c)*x^2*(b*x^2+a*x)^(3/2)/d^2+1 
/6*x*(b*x^2+a*x)^(5/2)/d+1/512*(-5*a^6*d^6-12*a^5*b*c*d^5-40*a^4*b^2*c^2*d 
^4-320*a^3*b^3*c^3*d^3+1920*a^2*b^4*c^4*d^2-2560*a*b^5*c^5*d+1024*b^6*c^6) 
*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(7/2)/d^7-2*c^(7/2)*(-a*d+b*c)^(5/ 
2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/d^7
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.85 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.42 \[ \int \frac {x \left (a x+b x^2\right )^{5/2}}{c+d x} \, dx=\frac {(x (a+b x))^{5/2} \left (\sqrt {b} d \sqrt {x} \sqrt {a+b x} \left (75 a^5 d^5+10 a^4 b d^4 (18 c-5 d x)+40 a^3 b^2 d^3 \left (15 c^2-3 c d x+d^2 x^2\right )+16 a^2 b^3 d^2 \left (-660 c^3+295 c^2 d x-186 c d^2 x^2+135 d^3 x^3\right )+64 a b^4 d \left (270 c^4-130 c^3 d x+85 c^2 d^2 x^2-63 c d^3 x^3+50 d^4 x^4\right )-128 b^5 \left (60 c^5-30 c^4 d x+20 c^3 d^2 x^2-15 c^2 d^3 x^3+12 c d^4 x^4-10 d^5 x^5\right )\right )+15360 b^{5/2} c^{5/2} (b c-a d)^2 \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+15360 b^{5/2} c^{5/2} (b c-a d)^2 \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+30 \left (1024 b^6 c^6-2560 a b^5 c^5 d+1920 a^2 b^4 c^4 d^2-320 a^3 b^3 c^3 d^3-40 a^4 b^2 c^2 d^4-12 a^5 b c d^5-5 a^6 d^6\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{7680 b^{7/2} d^7 x^{5/2} (a+b x)^{5/2}} \] Input:

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

Output:

((x*(a + b*x))^(5/2)*(Sqrt[b]*d*Sqrt[x]*Sqrt[a + b*x]*(75*a^5*d^5 + 10*a^4 
*b*d^4*(18*c - 5*d*x) + 40*a^3*b^2*d^3*(15*c^2 - 3*c*d*x + d^2*x^2) + 16*a 
^2*b^3*d^2*(-660*c^3 + 295*c^2*d*x - 186*c*d^2*x^2 + 135*d^3*x^3) + 64*a*b 
^4*d*(270*c^4 - 130*c^3*d*x + 85*c^2*d^2*x^2 - 63*c*d^3*x^3 + 50*d^4*x^4) 
- 128*b^5*(60*c^5 - 30*c^4*d*x + 20*c^3*d^2*x^2 - 15*c^2*d^3*x^3 + 12*c*d^ 
4*x^4 - 10*d^5*x^5)) + 15360*b^(5/2)*c^(5/2)*(b*c - a*d)^2*(b*c - a*d - I* 
Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[ 
d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sq 
rt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sqrt[a] + Sqrt[a + b*x]))] + 15360*b^(5 
/2)*c^(5/2)*(b*c - a*d)^2*(b*c - a*d + I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])* 
Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[ 
-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]* 
(-Sqrt[a] + Sqrt[a + b*x]))] + 30*(1024*b^6*c^6 - 2560*a*b^5*c^5*d + 1920* 
a^2*b^4*c^4*d^2 - 320*a^3*b^3*c^3*d^3 - 40*a^4*b^2*c^2*d^4 - 12*a^5*b*c*d^ 
5 - 5*a^6*d^6)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(76 
80*b^(7/2)*d^7*x^(5/2)*(a + b*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 488, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1231, 27, 1231, 27, 1231, 27, 1269, 1091, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {\int -\frac {\left (a c (12 b c-5 a d)+\left (24 b^2 c^2-12 a b d c-5 a^2 d^2\right ) x\right ) \left (b x^2+a x\right )^{3/2}}{2 (c+d x)}dx}{12 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (a c (12 b c-5 a d)+\left (24 b^2 c^2-12 a b d c-5 a^2 d^2\right ) x\right ) \left (b x^2+a x\right )^{3/2}}{c+d x}dx}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {-\frac {\int -\frac {3 \left (a c \left (64 b^3 c^3-88 a b^2 d c^2+12 a^2 b d^2 c+5 a^3 d^3\right )+\left (128 b^4 c^4-192 a b^3 d c^3+40 a^2 b^2 d^2 c^2+12 a^3 b d^3 c+5 a^4 d^4\right ) x\right ) \sqrt {b x^2+a x}}{2 (c+d x)}dx}{8 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {\left (a c \left (64 b^3 c^3-88 a b^2 d c^2+12 a^2 b d^2 c+5 a^3 d^3\right )+\left (128 b^4 c^4-192 a b^3 d c^3+40 a^2 b^2 d^2 c^2+12 a^3 b d^3 c+5 a^4 d^4\right ) x\right ) \sqrt {b x^2+a x}}{c+d x}dx}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {3 \left (-\frac {\int -\frac {a c \left (512 b^5 c^5-1152 a b^4 d c^4+704 a^2 b^3 d^2 c^3-40 a^3 b^2 d^3 c^2-12 a^4 b d^4 c-5 a^5 d^5\right )+\left (1024 b^6 c^6-2560 a b^5 d c^5+1920 a^2 b^4 d^2 c^4-320 a^3 b^3 d^3 c^3-40 a^4 b^2 d^4 c^2-12 a^5 b d^5 c-5 a^6 d^6\right ) x}{2 (c+d x) \sqrt {b x^2+a x}}dx}{4 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {a c \left (512 b^5 c^5-1152 a b^4 d c^4+704 a^2 b^3 d^2 c^3-40 a^3 b^2 d^3 c^2-12 a^4 b d^4 c-5 a^5 d^5\right )+\left (1024 b^6 c^6-2560 a b^5 d c^5+1920 a^2 b^4 d^2 c^4-320 a^3 b^3 d^3 c^3-40 a^4 b^2 d^4 c^2-12 a^5 b d^5 c-5 a^6 d^6\right ) x}{(c+d x) \sqrt {b x^2+a x}}dx}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {\left (-5 a^6 d^6-12 a^5 b c d^5-40 a^4 b^2 c^2 d^4-320 a^3 b^3 c^3 d^3+1920 a^2 b^4 c^4 d^2-2560 a b^5 c^5 d+1024 b^6 c^6\right ) \int \frac {1}{\sqrt {b x^2+a x}}dx}{d}-\frac {1024 b^3 c^4 (b c-a d)^3 \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 \left (-5 a^6 d^6-12 a^5 b c d^5-40 a^4 b^2 c^2 d^4-320 a^3 b^3 c^3 d^3+1920 a^2 b^4 c^4 d^2-2560 a b^5 c^5 d+1024 b^6 c^6\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{d}-\frac {1024 b^3 c^4 (b c-a d)^3 \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (-5 a^6 d^6-12 a^5 b c d^5-40 a^4 b^2 c^2 d^4-320 a^3 b^3 c^3 d^3+1920 a^2 b^4 c^4 d^2-2560 a b^5 c^5 d+1024 b^6 c^6\right )}{\sqrt {b} d}-\frac {1024 b^3 c^4 (b c-a d)^3 \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2048 b^3 c^4 (b c-a d)^3 \int \frac {1}{4 c (b c-a d)-\frac {(a c+(2 b c-a d) x)^2}{b x^2+a x}}d\left (-\frac {a c+(2 b c-a d) x}{\sqrt {b x^2+a x}}\right )}{d}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (-5 a^6 d^6-12 a^5 b c d^5-40 a^4 b^2 c^2 d^4-320 a^3 b^3 c^3 d^3+1920 a^2 b^4 c^4 d^2-2560 a b^5 c^5 d+1024 b^6 c^6\right )}{\sqrt {b} d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (-5 a^6 d^6-12 a^5 b c d^5-40 a^4 b^2 c^2 d^4-320 a^3 b^3 c^3 d^3+1920 a^2 b^4 c^4 d^2-2560 a b^5 c^5 d+1024 b^6 c^6\right )}{\sqrt {b} d}-\frac {1024 b^3 c^{7/2} (b c-a d)^{5/2} \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (-5 a^5 d^5-12 a^4 b c d^4-40 a^3 b^2 c^2 d^3+704 a^2 b^3 c^3 d^2-2 b d x \left (5 a^4 d^4+12 a^3 b c d^3+40 a^2 b^2 c^2 d^2-192 a b^3 c^3 d+128 b^4 c^4\right )-1152 a b^4 c^4 d+512 b^5 c^5\right )}{4 b d^2}\right )}{16 b d^2}-\frac {\left (a x+b x^2\right )^{3/2} \left (5 a^3 d^3-2 b d x \left (-5 a^2 d^2-12 a b c d+24 b^2 c^2\right )+12 a^2 b c d^2-88 a b^2 c^2 d+64 b^3 c^3\right )}{8 b d^2}}{24 b d^2}-\frac {\left (a x+b x^2\right )^{5/2} (-5 a d+12 b c-10 b d x)}{60 b d^2}\)

Input:

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

Output:

-1/60*((12*b*c - 5*a*d - 10*b*d*x)*(a*x + b*x^2)^(5/2))/(b*d^2) + (-1/8*(( 
64*b^3*c^3 - 88*a*b^2*c^2*d + 12*a^2*b*c*d^2 + 5*a^3*d^3 - 2*b*d*(24*b^2*c 
^2 - 12*a*b*c*d - 5*a^2*d^2)*x)*(a*x + b*x^2)^(3/2))/(b*d^2) + (3*(-1/4*(( 
512*b^5*c^5 - 1152*a*b^4*c^4*d + 704*a^2*b^3*c^3*d^2 - 40*a^3*b^2*c^2*d^3 
- 12*a^4*b*c*d^4 - 5*a^5*d^5 - 2*b*d*(128*b^4*c^4 - 192*a*b^3*c^3*d + 40*a 
^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + 5*a^4*d^4)*x)*Sqrt[a*x + b*x^2])/(b*d^2) 
 + ((2*(1024*b^6*c^6 - 2560*a*b^5*c^5*d + 1920*a^2*b^4*c^4*d^2 - 320*a^3*b 
^3*c^3*d^3 - 40*a^4*b^2*c^2*d^4 - 12*a^5*b*c*d^5 - 5*a^6*d^6)*ArcTanh[(Sqr 
t[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*d) - (1024*b^3*c^(7/2)*(b*c - a*d)^(5 
/2)*ArcTanh[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + 
b*x^2])])/d)/(8*b*d^2)))/(16*b*d^2))/(24*b*d^2)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {\left (1280 b^{5} d^{5} x^{5}+3200 a \,b^{4} d^{5} x^{4}-1536 b^{5} c \,d^{4} x^{4}+2160 a^{2} b^{3} d^{5} x^{3}-4032 a \,b^{4} c \,d^{4} x^{3}+1920 b^{5} c^{2} d^{3} x^{3}+40 a^{3} b^{2} d^{5} x^{2}-2976 a^{2} b^{3} c \,d^{4} x^{2}+5440 a \,b^{4} c^{2} d^{3} x^{2}-2560 b^{5} c^{3} d^{2} x^{2}-50 a^{4} b \,d^{5} x -120 a^{3} b^{2} c \,d^{4} x +4720 a^{2} b^{3} c^{2} d^{3} x -8320 a \,b^{4} c^{3} d^{2} x +3840 b^{5} c^{4} d x +75 a^{5} d^{5}+180 a^{4} b c \,d^{4}+600 a^{3} b^{2} c^{2} d^{3}-10560 a^{2} b^{3} c^{3} d^{2}+17280 a \,b^{4} c^{4} d -7680 b^{5} c^{5}\right ) x \left (b x +a \right )}{7680 b^{3} d^{6} \sqrt {x \left (b x +a \right )}}-\frac {\frac {\left (5 a^{6} d^{6}+12 a^{5} b c \,d^{5}+40 a^{4} b^{2} c^{2} d^{4}+320 a^{3} b^{3} c^{3} d^{3}-1920 a^{2} b^{4} c^{4} d^{2}+2560 a \,b^{5} c^{5} d -1024 b^{6} c^{6}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {1024 c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3} \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{1024 d^{6} b^{3}}\)	\(577\)
default	\(\text {Expression too large to display}\)	\(1050\)

Input:

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

Output:

1/7680/b^3*(1280*b^5*d^5*x^5+3200*a*b^4*d^5*x^4-1536*b^5*c*d^4*x^4+2160*a^ 
2*b^3*d^5*x^3-4032*a*b^4*c*d^4*x^3+1920*b^5*c^2*d^3*x^3+40*a^3*b^2*d^5*x^2 
-2976*a^2*b^3*c*d^4*x^2+5440*a*b^4*c^2*d^3*x^2-2560*b^5*c^3*d^2*x^2-50*a^4 
*b*d^5*x-120*a^3*b^2*c*d^4*x+4720*a^2*b^3*c^2*d^3*x-8320*a*b^4*c^3*d^2*x+3 
840*b^5*c^4*d*x+75*a^5*d^5+180*a^4*b*c*d^4+600*a^3*b^2*c^2*d^3-10560*a^2*b 
^3*c^3*d^2+17280*a*b^4*c^4*d-7680*b^5*c^5)*x*(b*x+a)/d^6/(x*(b*x+a))^(1/2) 
-1/1024/d^6/b^3*((5*a^6*d^6+12*a^5*b*c*d^5+40*a^4*b^2*c^2*d^4+320*a^3*b^3* 
c^3*d^3-1920*a^2*b^4*c^4*d^2+2560*a*b^5*c^5*d-1024*b^6*c^6)/d*ln((1/2*a+b* 
x)/b^(1/2)+(b*x^2+a*x)^(1/2))/b^(1/2)+1024*c^4*(a^3*d^3-3*a^2*b*c*d^2+3*a* 
b^2*c^2*d-b^3*c^3)*b^3/d^2/(-c*(a*d-b*c)/d^2)^(1/2)*ln((-2*c*(a*d-b*c)/d^2 
+(a*d-2*b*c)/d*(x+c/d)+2*(-c*(a*d-b*c)/d^2)^(1/2)*(b*(x+c/d)^2+(a*d-2*b*c) 
/d*(x+c/d)-c*(a*d-b*c)/d^2)^(1/2))/(x+c/d)))
 

Fricas [A] (verification not implemented)

Time = 3.42 (sec) , antiderivative size = 1920, normalized size of antiderivative = 3.89 \[ \int \frac {x \left (a x+b x^2\right )^{5/2}}{c+d x} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/15360*(15*(1024*b^6*c^6 - 2560*a*b^5*c^5*d + 1920*a^2*b^4*c^4*d^2 - 32 
0*a^3*b^3*c^3*d^3 - 40*a^4*b^2*c^2*d^4 - 12*a^5*b*c*d^5 - 5*a^6*d^6)*sqrt( 
b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 15360*(b^6*c^5 - 2*a*b^5 
*c^4*d + a^2*b^4*c^3*d^2)*sqrt(b*c^2 - a*c*d)*log((a*c + (2*b*c - a*d)*x - 
 2*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) - 2*(1280*b^6*d^6*x^5 
 - 7680*b^6*c^5*d + 17280*a*b^5*c^4*d^2 - 10560*a^2*b^4*c^3*d^3 + 600*a^3* 
b^3*c^2*d^4 + 180*a^4*b^2*c*d^5 + 75*a^5*b*d^6 - 128*(12*b^6*c*d^5 - 25*a* 
b^5*d^6)*x^4 + 48*(40*b^6*c^2*d^4 - 84*a*b^5*c*d^5 + 45*a^2*b^4*d^6)*x^3 - 
 8*(320*b^6*c^3*d^3 - 680*a*b^5*c^2*d^4 + 372*a^2*b^4*c*d^5 - 5*a^3*b^3*d^ 
6)*x^2 + 10*(384*b^6*c^4*d^2 - 832*a*b^5*c^3*d^3 + 472*a^2*b^4*c^2*d^4 - 1 
2*a^3*b^3*c*d^5 - 5*a^4*b^2*d^6)*x)*sqrt(b*x^2 + a*x))/(b^4*d^7), 1/15360* 
(30720*(b^6*c^5 - 2*a*b^5*c^4*d + a^2*b^4*c^3*d^2)*sqrt(-b*c^2 + a*c*d)*ar 
ctan(sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a*x)/(b*c*x + a*c)) - 15*(1024*b^6* 
c^6 - 2560*a*b^5*c^5*d + 1920*a^2*b^4*c^4*d^2 - 320*a^3*b^3*c^3*d^3 - 40*a 
^4*b^2*c^2*d^4 - 12*a^5*b*c*d^5 - 5*a^6*d^6)*sqrt(b)*log(2*b*x + a - 2*sqr 
t(b*x^2 + a*x)*sqrt(b)) + 2*(1280*b^6*d^6*x^5 - 7680*b^6*c^5*d + 17280*a*b 
^5*c^4*d^2 - 10560*a^2*b^4*c^3*d^3 + 600*a^3*b^3*c^2*d^4 + 180*a^4*b^2*c*d 
^5 + 75*a^5*b*d^6 - 128*(12*b^6*c*d^5 - 25*a*b^5*d^6)*x^4 + 48*(40*b^6*c^2 
*d^4 - 84*a*b^5*c*d^5 + 45*a^2*b^4*d^6)*x^3 - 8*(320*b^6*c^3*d^3 - 680*a*b 
^5*c^2*d^4 + 372*a^2*b^4*c*d^5 - 5*a^3*b^3*d^6)*x^2 + 10*(384*b^6*c^4*d...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(b*x^2+a*x)^(5/2)/(d*x+c),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-2*b*c>0)', see `assume?` for 
 more deta
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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