\(\int \frac {(f+g x) (a+b x+c x^2)^{3/2}}{(d+e x)^5} \, dx\) [888]

Optimal result

Integrand size = 27, antiderivative size = 526 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {c g \sqrt {a+b x+c x^2}}{e^4 (d+e x)}+\frac {\left (8 b e \left (3 c d^2+a e^2\right ) g-b^2 e^2 (3 e f+5 d g)-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {g \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)^3}+\frac {(e f-d g) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 e \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac {c^{3/2} g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^5}+\frac {\left (8 b^3 e^3 \left (5 c d^2-a e^2\right ) g+32 b c e \left (10 c^2 d^4+15 a c d^2 e^2+3 a^2 e^4\right ) g+b^4 e^4 (3 e f+5 d g)-16 c^2 \left (8 c^2 d^5 g+20 a c d^3 e^2 g-3 a^2 e^4 (e f-5 d g)\right )-24 b^2 c e^2 \left (10 c d^3 g+a e^2 (e f+5 d g)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 e^5 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-c*g*(c*x^2+b*x+a)^(1/2)/e^4/(e*x+d)+1/64*(8*b*e*(a*e^2+3*c*d^2)*g-b^2*e^2 
*(5*d*g+3*e*f)-4*c*(4*c*d^3*g-a*e^2*(-7*d*g+3*e*f)))*(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*g*(c*x^2 
+b*x+a)^(3/2)/e^2/(e*x+d)^3+1/8*(-d*g+e*f)*(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)*g*arctanh(1/2*(2*c 
*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^5+1/128*(8*b^3*e^3*(-a*e^2+5*c*d^2)*g 
+32*b*c*e*(3*a^2*e^4+15*a*c*d^2*e^2+10*c^2*d^4)*g+b^4*e^4*(5*d*g+3*e*f)-16 
*c^2*(8*c^2*d^5*g+20*a*c*d^3*e^2*g-3*a^2*e^4*(-5*d*g+e*f))-24*b^2*c*e^2*(1 
0*c*d^3*g+a*e^2*(5*d*g+e*f)))*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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(8311\) vs. \(2(526)=1052\).

Time = 16.55 (sec) , antiderivative size = 8311, normalized size of antiderivative = 15.80 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.39, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int -\frac {\left (-e (3 e f+5 d g) b^2+8 \left (c d^2+a e^2\right ) g b+12 a c e (e f-d g)+16 c \left (c d^2-b e d+a e^2\right ) g x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^3}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (-e (3 e f+5 d g) b^2+8 \left (c d^2+a e^2\right ) g b+12 a c e (e f-d g)+16 c \left (c d^2-b e d+a e^2\right ) g x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {\int -\frac {e^3 (3 e f+5 d g) b^4+8 e^2 \left (5 c d^2-a e^2\right ) g b^3-8 c e \left (14 c g d^3+3 a e^2 (e f+5 d g)\right ) b^2+32 c \left (2 c^2 d^4+7 a c e^2 d^2+3 a^2 e^4\right ) g b-16 a c^2 e \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )+128 c^2 \left (c d^2-b e d+a e^2\right )^2 g x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {e^3 (3 e f+5 d g) b^4+8 e^2 \left (5 c d^2-a e^2\right ) g b^3-8 c e \left (14 c g d^3+3 a e^2 (e f+5 d g)\right ) b^2+32 c \left (2 c^2 d^4+7 a c e^2 d^2+3 a^2 e^4\right ) g b-16 a c^2 e \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )+128 c^2 \left (c d^2-b e d+a e^2\right )^2 g x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {\left (32 b c e g \left (3 a^2 e^4+15 a c d^2 e^2+10 c^2 d^4\right )-16 c^2 \left (-3 a^2 e^4 (e f-5 d g)+20 a c d^3 e^2 g+8 c^2 d^5 g\right )+8 b^3 e^3 g \left (5 c d^2-a e^2\right )-24 b^2 c e^2 \left (a e^2 (5 d g+e f)+10 c d^3 g\right )+b^4 e^4 (5 d g+3 e f)\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {128 c^2 g \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\left (32 b c e g \left (3 a^2 e^4+15 a c d^2 e^2+10 c^2 d^4\right )-16 c^2 \left (-3 a^2 e^4 (e f-5 d g)+20 a c d^3 e^2 g+8 c^2 d^5 g\right )+8 b^3 e^3 g \left (5 c d^2-a e^2\right )-24 b^2 c e^2 \left (a e^2 (5 d g+e f)+10 c d^3 g\right )+b^4 e^4 (5 d g+3 e f)\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {256 c^2 g \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (32 b c e g \left (3 a^2 e^4+15 a c d^2 e^2+10 c^2 d^4\right )-16 c^2 \left (-3 a^2 e^4 (e f-5 d g)+20 a c d^3 e^2 g+8 c^2 d^5 g\right )+8 b^3 e^3 g \left (5 c d^2-a e^2\right )-24 b^2 c e^2 \left (a e^2 (5 d g+e f)+10 c d^3 g\right )+b^4 e^4 (5 d g+3 e f)\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {128 c^{3/2} g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {128 c^{3/2} g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}-\frac {2 \left (32 b c e g \left (3 a^2 e^4+15 a c d^2 e^2+10 c^2 d^4\right )-16 c^2 \left (-3 a^2 e^4 (e f-5 d g)+20 a c d^3 e^2 g+8 c^2 d^5 g\right )+8 b^3 e^3 g \left (5 c d^2-a e^2\right )-24 b^2 c e^2 \left (a e^2 (5 d g+e f)+10 c d^3 g\right )+b^4 e^4 (5 d g+3 e f)\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right ) \left (32 b c e g \left (3 a^2 e^4+15 a c d^2 e^2+10 c^2 d^4\right )-16 c^2 \left (-3 a^2 e^4 (e f-5 d g)+20 a c d^3 e^2 g+8 c^2 d^5 g\right )+8 b^3 e^3 g \left (5 c d^2-a e^2\right )-24 b^2 c e^2 \left (a e^2 (5 d g+e f)+10 c d^3 g\right )+b^4 e^4 (5 d g+3 e f)\right )}{e \sqrt {a e^2-b d e+c d^2}}+\frac {128 c^{3/2} g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b \left (-8 a^2 e^5 g+2 a c d e^3 (13 d g+3 e f)+56 c^2 d^4 e g\right )+8 c \left (a^2 e^4 (d g+3 e f)+12 a c d^3 e^2 g+8 c^2 d^5 g\right )+2 b^2 \left (20 c d^3 e^2 g-3 a e^4 (3 d g+e f)\right )+e x \left (64 c g \left (c d^2-e (b d-a e)\right )^2-(2 c d-b e) \left (8 b e g \left (a e^2+3 c d^2\right )-4 c \left (4 c d^3 g-a e^2 (3 e f-7 d g)\right )-b^2 e^2 (5 d g+3 e f)\right )\right )+b^3 d e^3 (5 d g+3 e f)\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-e x (2 c d (3 e f-7 d g)-e (8 a e g-11 b d g+3 b e f))+2 a e^2 (d g+3 e f)-b d e (5 d g+3 e f)+8 c d^3 g\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-1/24*((8*c*d^3*g + 2*a*e^2*(3*e*f + d*g) - b*d*e*(3*e*f + 5*d*g) - e*(2*c 
*d*(3*e*f - 7*d*g) - e*(3*b*e*f - 11*b*d*g + 8*a*e*g))*x)*(a + b*x + c*x^2 
)^(3/2))/(e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + (-1/4*((b^3*d*e^3*(3* 
e*f + 5*d*g) + 8*c*(8*c^2*d^5*g + 12*a*c*d^3*e^2*g + a^2*e^4*(3*e*f + d*g) 
) + 2*b^2*(20*c*d^3*e^2*g - 3*a*e^4*(e*f + 3*d*g)) - 2*b*(56*c^2*d^4*e*g - 
 8*a^2*e^5*g + 2*a*c*d*e^3*(3*e*f + 13*d*g)) + e*(64*c*(c*d^2 - e*(b*d - a 
*e))^2*g - (2*c*d - b*e)*(8*b*e*(3*c*d^2 + a*e^2)*g - b^2*e^2*(3*e*f + 5*d 
*g) - 4*c*(4*c*d^3*g - a*e^2*(3*e*f - 7*d*g))))*x)*Sqrt[a + b*x + c*x^2])/ 
(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*g*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e + (( 
8*b^3*e^3*(5*c*d^2 - a*e^2)*g + 32*b*c*e*(10*c^2*d^4 + 15*a*c*d^2*e^2 + 3* 
a^2*e^4)*g + b^4*e^4*(3*e*f + 5*d*g) - 16*c^2*(8*c^2*d^5*g + 20*a*c*d^3*e^ 
2*g - 3*a^2*e^4*(e*f - 5*d*g)) - 24*b^2*c*e^2*(10*c*d^3*g + a*e^2*(e*f + 5 
*d*g)))*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. \(8168\) vs. \(2(494)=988\).

Time = 3.33 (sec) , antiderivative size = 8169, normalized size of antiderivative = 15.53

method	result	size
default	\(\text {Expression too large to display}\)	\(8169\)

Input:

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 11.92 (sec) , antiderivative size = 13702, normalized size of antiderivative = 26.05 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(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*g - 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 
**3*e**6*g*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*g*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*g*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*g*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*g - 144*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**4*e**5*f + 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**4*e**5*g*x - 576*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*f*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...