\(\int \frac {x (a+b x+c x^2)^{3/2}}{(d+e x)^5} \, dx\) [82]

Optimal result

Integrand size = 23, antiderivative size = 440 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {c \sqrt {a+b x+c x^2}}{e^4 (d+e x)}-\frac {\left (16 c^2 d^3+b e^2 (5 b d-8 a e)-4 c d e (6 b d-7 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac {d (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 e \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^5}-\frac {\left (128 c^4 d^5-b^3 e^4 (5 b d-8 a e)-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2-8 b c e^3 \left (5 b^2 d^2-15 a b d e+12 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 e^5 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-c*(c*x^2+b*x+a)^(1/2)/e^4/(e*x+d)-1/64*(16*c^2*d^3+b*e^2*(-8*a*e+5*b*d)-4 
*c*d*e*(-7*a*e+6*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e^3/ 
(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2-1/3*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)^3-1/8* 
d*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/e/(a*e^2-b*d*e+c*d^2)/(e* 
x+d)^4+c^(3/2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^5-1/12 
8*(128*c^4*d^5-b^3*e^4*(-8*a*e+5*b*d)-320*c^3*d^3*e*(-a*e+b*d)+240*c^2*d*e 
^2*(-a*e+b*d)^2-8*b*c*e^3*(12*a^2*e^2-15*a*b*d*e+5*b^2*d^2))*arctanh(1/2*( 
b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e 
^5/(a*e^2-b*d*e+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 14.36 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {-\frac {2 e \sqrt {a+x (b+c x)} \left (16 c^3 d^4 \left (12 d^3+42 d^2 e x+52 d e^2 x^2+25 e^3 x^3\right )+e^3 \left (16 a^3 e^3 (d+4 e x)-8 a^2 b e^2 \left (d^2+5 d e x-14 e^2 x^2\right )+b^3 d \left (15 d^3+55 d^2 e x+73 d e^2 x^2-15 e^3 x^3\right )-2 a b^2 e \left (7 d^3+26 d^2 e x+79 d e^2 x^2-12 e^3 x^3\right )\right )-8 c^2 d^2 e \left (-a e \left (44 d^3+155 d^2 e x+184 d e^2 x^2+91 e^3 x^3\right )+b d \left (42 d^3+148 d^2 e x+185 d e^2 x^2+97 e^3 x^3\right )\right )+2 c e^2 \left (4 a^2 e^2 \left (13 d^3+52 d^2 e x+53 d e^2 x^2+32 e^3 x^3\right )-2 a b d e \left (65 d^3+237 d^2 e x+267 d e^2 x^2+167 e^3 x^3\right )+b^2 d^2 \left (60 d^3+215 d^2 e x+274 d e^2 x^2+191 e^3 x^3\right )\right )\right )}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)^4}+384 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+\frac {3 \left (128 c^4 d^5-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2+b^3 e^4 (-5 b d+8 a e)-8 b c e^3 \left (5 b^2 d^2-15 a b d e+12 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}}}{384 e^5} \] Input:

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

Output:

((-2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*d^4*(12*d^3 + 42*d^2*e*x + 52*d*e^2*x 
^2 + 25*e^3*x^3) + e^3*(16*a^3*e^3*(d + 4*e*x) - 8*a^2*b*e^2*(d^2 + 5*d*e* 
x - 14*e^2*x^2) + b^3*d*(15*d^3 + 55*d^2*e*x + 73*d*e^2*x^2 - 15*e^3*x^3) 
- 2*a*b^2*e*(7*d^3 + 26*d^2*e*x + 79*d*e^2*x^2 - 12*e^3*x^3)) - 8*c^2*d^2* 
e*(-(a*e*(44*d^3 + 155*d^2*e*x + 184*d*e^2*x^2 + 91*e^3*x^3)) + b*d*(42*d^ 
3 + 148*d^2*e*x + 185*d*e^2*x^2 + 97*e^3*x^3)) + 2*c*e^2*(4*a^2*e^2*(13*d^ 
3 + 52*d^2*e*x + 53*d*e^2*x^2 + 32*e^3*x^3) - 2*a*b*d*e*(65*d^3 + 237*d^2* 
e*x + 267*d*e^2*x^2 + 167*e^3*x^3) + b^2*d^2*(60*d^3 + 215*d^2*e*x + 274*d 
*e^2*x^2 + 191*e^3*x^3))))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^4) + 38 
4*c^(3/2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (3*(128 
*c^4*d^5 - 320*c^3*d^3*e*(b*d - a*e) + 240*c^2*d*e^2*(b*d - a*e)^2 + b^3*e 
^4*(-5*b*d + 8*a*e) - 8*b*c*e^3*(5*b^2*d^2 - 15*a*b*d*e + 12*a^2*e^2))*Arc 
Tanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]* 
Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2))/(384*e^5)
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.32, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1229, 27, 1229, 27, 1269, 1092, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int \frac {\left (5 d e b^2-8 \left (c d^2+a e^2\right ) b+12 a c d e-16 c \left (c d^2-b e d+a e^2\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^3}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\left (5 d e b^2-8 \left (c d^2+a e^2\right ) b+12 a c d e-16 c \left (c d^2-b e d+a e^2\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {5 d e^3 b^4+8 \left (5 c d^2 e^2-a e^4\right ) b^3-8 c d e \left (14 c d^2+15 a e^2\right ) b^2+32 c \left (2 c^2 d^4+7 a c e^2 d^2+3 a^2 e^4\right ) b-16 a c^2 d e \left (4 c d^2+7 a e^2\right )+128 c^2 \left (c d^2-b e d+a e^2\right )^2 x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {5 d e^3 b^4+8 \left (5 c d^2 e^2-a e^4\right ) b^3-8 c d e \left (14 c d^2+15 a e^2\right ) b^2+32 c \left (2 c^2 d^4+7 a c e^2 d^2+3 a^2 e^4\right ) b-16 a c^2 d e \left (4 c d^2+7 a e^2\right )+128 c^2 \left (c d^2-b e d+a e^2\right )^2 x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {128 c^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-8 b c e^3 \left (12 a^2 e^2-15 a b d e+5 b^2 d^2\right )-b^3 e^4 (5 b d-8 a e)-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2+128 c^4 d^5\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {256 c^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {\left (-8 b c e^3 \left (12 a^2 e^2-15 a b d e+5 b^2 d^2\right )-b^3 e^4 (5 b d-8 a e)-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2+128 c^4 d^5\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {128 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}-\frac {\left (-8 b c e^3 \left (12 a^2 e^2-15 a b d e+5 b^2 d^2\right )-b^3 e^4 (5 b d-8 a e)-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2+128 c^4 d^5\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {2 \left (-8 b c e^3 \left (12 a^2 e^2-15 a b d e+5 b^2 d^2\right )-b^3 e^4 (5 b d-8 a e)-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2+128 c^4 d^5\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {128 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\sqrt {a+b x+c x^2} \left (4 c d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+b e^3 \left (16 a^2 e^2-18 a b d e+5 b^2 d^2\right )-16 c^2 d^3 e (7 b d-6 a e)+e x \left ((2 c d-b e) \left (-4 c d e (6 b d-7 a e)+b e^2 (5 b d-8 a e)+16 c^2 d^3\right )+64 c \left (c d^2-e (b d-a e)\right )^2\right )+64 c^3 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {128 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}-\frac {\left (-8 b c e^3 \left (12 a^2 e^2-15 a b d e+5 b^2 d^2\right )-b^3 e^4 (5 b d-8 a e)-320 c^3 d^3 e (b d-a e)+240 c^2 d e^2 (b d-a e)^2+128 c^4 d^5\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (14 c d^2-e (11 b d-8 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-1/24*((d*(8*c*d^2 - e*(5*b*d - 2*a*e)) + e*(14*c*d^2 - e*(11*b*d - 8*a*e) 
)*x)*(a + b*x + c*x^2)^(3/2))/(e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - 
(((64*c^3*d^5 - 16*c^2*d^3*e*(7*b*d - 6*a*e) + 4*c*d*e^2*(10*b^2*d^2 - 13* 
a*b*d*e + 2*a^2*e^2) + b*e^3*(5*b^2*d^2 - 18*a*b*d*e + 16*a^2*e^2) + e*((2 
*c*d - b*e)*(16*c^2*d^3 + b*e^2*(5*b*d - 8*a*e) - 4*c*d*e*(6*b*d - 7*a*e)) 
 + 64*c*(c*d^2 - e*(b*d - a*e))^2)*x)*Sqrt[a + b*x + c*x^2])/(4*e^2*(c*d^2 
 - b*d*e + a*e^2)*(d + e*x)^2) - ((128*c^(3/2)*(c*d^2 - b*d*e + a*e^2)^2*A 
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e - ((128*c^4*d^5 - 
 b^3*e^4*(5*b*d - 8*a*e) - 320*c^3*d^3*e*(b*d - a*e) + 240*c^2*d*e^2*(b*d 
- a*e)^2 - 8*b*c*e^3*(5*b^2*d^2 - 15*a*b*d*e + 12*a^2*e^2))*ArcTanh[(b*d - 
 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c* 
x^2])])/(e*Sqrt[c*d^2 - b*d*e + a*e^2]))/(8*e^2*(c*d^2 - b*d*e + a*e^2)))/ 
(16*e^2*(c*d^2 - b*d*e + a*e^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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, 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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8160\) vs. \(2(408)=816\).

Time = 1.69 (sec) , antiderivative size = 8161, normalized size of antiderivative = 18.55

method	result	size
default	\(\text {Expression too large to display}\)	\(8161\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(c*x^2+b*x+a)^(3/2)/(e*x+d)^5,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*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 17.56 (sec) , antiderivative size = 10533, normalized size of antiderivative = 23.94 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 288*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt 
(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c*d**4*e 
**5 - 1152*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*s 
qrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c*d** 
3*e**6*x - 1728*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x* 
*2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b* 
c*d**2*e**7*x**2 - 1152*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b* 
x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x) 
*a**2*b*c*d*e**8*x**3 - 288*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a 
+ b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c* 
d*x)*a**2*b*c*e**9*x**4 + 720*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt( 
a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2* 
c*d*x)*a**2*c**2*d**5*e**4 + 2880*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*s 
qrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x 
+ 2*c*d*x)*a**2*c**2*d**4*e**5*x + 4320*sqrt(a*e**2 - b*d*e + c*d**2)*log( 
 - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - 
b*e*x + 2*c*d*x)*a**2*c**2*d**3*e**6*x**2 + 2880*sqrt(a*e**2 - b*d*e + c*d 
**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e 
 + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**2*e**7*x**3 + 720*sqrt(a*e**2 - b*d 
*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d*...