\(\int \frac {(d+e x)^{9/2}}{(a+b x+c x^2)^{5/2}} \, dx\) [669]

Optimal result

Integrand size = 24, antiderivative size = 804 \[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{7/2} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2} \left (11 b^2 d e-12 a c d e-8 b \left (c d^2+a e^2\right )-\left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {8 e (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}-\frac {2 e \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{3 c \left (b^2-4 a c\right )^2}-\frac {\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \left (b^2-4 a c\right )^{3/2} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/3*(e*x+d)^(7/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^( 
3/2)-2/3*(e*x+d)^(5/2)*(11*b^2*d*e-12*a*c*d*e-8*b*(a*e^2+c*d^2)-(16*c^2*d^ 
2-3*b^2*e^2-4*c*e*(-7*a*e+4*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)-8/ 
3*e*(-b*e+2*c*d)*(2*c^2*d^2-b^2*e^2-2*c*e*(-3*a*e+b*d))*(e*x+d)^(1/2)*(c*x 
^2+b*x+a)^(1/2)/c^2/(-4*a*c+b^2)^2-2/3*e*(16*c^2*d^2-3*b^2*e^2-4*c*e*(-7*a 
*e+4*b*d))*(e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)^2-1/3*2^(1/2)* 
(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(-15*a*e+8*b*d)+b^2*c*e^3*(57*a*e+7*b*d) 
+3*c^2*e^2*(-28*a^2*e^2-20*a*b*d*e+3*b^2*d^2))*(e*x+d)^(1/2)*(-c*(c*x^2+b* 
x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1 
/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/ 
2))/c^3/(-4*a*c+b^2)^(3/2)/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1 
/2)/(c*x^2+b*x+a)^(1/2)+8/3*2^(1/2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(2*c^ 
2*d^2-b^2*e^2-2*c*e*(-3*a*e+b*d))*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2)) 
*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*c*x+b 
)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4 
*a*c+b^2)^(1/2))*e))^(1/2))/c^3/(-4*a*c+b^2)^(3/2)/(e*x+d)^(1/2)/(c*x^2+b* 
x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.77 (sec) , antiderivative size = 7889, normalized size of antiderivative = 9.81 \[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(d + e*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 1.46 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1164, 27, 1233, 27, 1236, 27, 1269, 1172, 321, 327}

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.

\(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {2 \int \frac {(d+e x)^{5/2} \left (8 c d^2-11 b e d+14 a e^2-3 e (2 c d-b e) x\right )}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(d+e x)^{5/2} \left (8 c d^2-e (11 b d-14 a e)-3 e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {\frac {2 \int \frac {3 e \sqrt {d+e x} \left (d e^2 b^3-\left (13 c d^2 e-3 a e^3\right ) b^2+4 c d \left (2 c d^2+5 a e^2\right ) b+4 a c e \left (c d^2-7 a e^2\right )+4 (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-3 a e^3\right )\right )+8 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (7 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3 e \int \frac {\sqrt {d+e x} \left (d e^2 b^3-\left (13 c d^2 e-3 a e^3\right ) b^2+4 c d \left (2 c d^2+5 a e^2\right ) b+4 a c e \left (c d^2-7 a e^2\right )+4 (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-3 a e^3\right )\right )+8 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (7 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1236

\(\displaystyle -\frac {\frac {3 e \left (\frac {2 \int -\frac {4 d e^3 b^4-\left (3 c d^2 e^2-4 a e^4\right ) b^3+3 c d e \left (5 c d^2-11 a e^2\right ) b^2-4 c \left (2 c^2 d^4+9 a c e^2 d^2+6 a^2 e^4\right ) b+4 a c^2 d e \left (c d^2+33 a e^2\right )-\left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {8 \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-3 a e^3\right )\right )+8 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (7 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3 e \left (\frac {8 \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )}{3 c}-\frac {\int \frac {4 d e^3 b^4-\left (3 c d^2 e^2-4 a e^4\right ) b^3+3 c d e \left (5 c d^2-11 a e^2\right ) b^2-4 c \left (2 c^2 d^4+9 a c e^2 d^2+6 a^2 e^4\right ) b+4 a c^2 d e \left (c d^2+33 a e^2\right )-\left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-3 a e^3\right )\right )+8 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (7 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {3 e \left (\frac {8 \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )}{3 c}-\frac {\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (6 a c e^2-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-3 a e^3\right )\right )+8 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (7 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{7/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2 (b d-2 a e+(2 c d-b e) x) (d+e x)^{7/2}}{3 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {3 e \left (\frac {8 (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d-b^2 e^2+6 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (-\left (\left (11 c d^2 e-3 a e^3\right ) b^2\right )+8 c d \left (c d^2+3 a e^2\right ) b-4 a c e \left (c d^2+7 a e^2\right )+(2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 (b d-2 a e+(2 c d-b e) x) (d+e x)^{7/2}}{3 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {3 e \left (\frac {8 (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d-b^2 e^2+6 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (-\left (\left (11 c d^2 e-3 a e^3\right ) b^2\right )+8 c d \left (c d^2+3 a e^2\right ) b-4 a c e \left (c d^2+7 a e^2\right )+(2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 (b d-2 a e+(2 c d-b e) x) (d+e x)^{7/2}}{3 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {3 e \left (\frac {8 (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d-b^2 e^2+6 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x)^{3/2} \left (-\left (\left (11 c d^2 e-3 a e^3\right ) b^2\right )+8 c d \left (c d^2+3 a e^2\right ) b-4 a c e \left (c d^2+7 a e^2\right )+(2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right )}\)

Input:

Int[(d + e*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]
 

Output:

(-2*(d + e*x)^(7/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + 
 b*x + c*x^2)^(3/2)) - ((-2*(d + e*x)^(3/2)*(8*b*c*d*(c*d^2 + 3*a*e^2) - 4 
*a*c*e*(c*d^2 + 7*a*e^2) - b^2*(11*c*d^2*e - 3*a*e^3) + (2*c*d - b*e)*(8*c 
^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 5*a*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + 
b*x + c*x^2]) + (3*e*((8*(2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 
 3*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) - (-((Sqrt[2]*Sqrt[b^2 
 - 4*a*c]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e 
^3*(7*b*d + 57*a*e) + c^2*(9*b^2*d^2*e^2 - 60*a*b*d*e^3 - 84*a^2*e^4))*Sqr 
t[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[S 
qrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[ 
b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e* 
x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2])) + (8*Sqrt 
[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(2*c^2*d^2 - 2 
*b*c*d*e - b^2*e^2 + 6*a*c*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 
- 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSi 
n[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq 
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x 
]*Sqrt[a + b*x + c*x^2]))/(3*c)))/(c*(b^2 - 4*a*c)))/(3*(b^2 - 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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1172

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

rule 1233

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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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. \(1835\) vs. \(2(732)=1464\).

Time = 9.06 (sec) , antiderivative size = 1836, normalized size of antiderivative = 2.28

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1836\)
default	\(\text {Expression too large to display}\)	\(26907\)

Input:

int((e*x+d)^(9/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*((2/3*(2*a 
^2*c^2*e^4-4*a*b^2*c*e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4-4*b^3*c 
*d*e^3+6*b^2*c^2*d^2*e^2-4*b*c^3*d^3*e+2*c^4*d^4)/c^5/(4*a*c-b^2)*x-2/3*(3 
*a^2*b*c*e^4-8*a^2*c^2*d*e^3-a*b^3*e^4+4*a*b^2*c*d*e^3-6*a*b*c^2*d^2*e^2+8 
*a*c^3*d^3*e-b*c^3*d^4)/c^5/(4*a*c-b^2))*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b* 
d*x+a*d)^(1/2)/(a/c+b/c*x+x^2)^2-2*(c*e*x+c*d)*(1/3/c^3*(36*a^2*c^2*e^4-33 
*a*b^2*c*e^4+60*a*b*c^2*d*e^3-60*a*c^3*d^2*e^2+5*b^4*e^4-7*b^3*c*d*e^3-9*b 
^2*c^2*d^2*e^2+32*b*c^3*d^3*e-16*c^4*d^4)/(4*a*c-b^2)^2*x-1/3*(36*a^2*b*c^ 
2*e^4-108*a^2*c^3*d*e^3-11*a*b^3*c*e^4+27*a*b^2*c^2*d*e^3+24*a*b*c^3*d^2*e 
^2+4*a*c^4*d^3*e+b^5*e^4-4*b^4*c*d*e^3+6*b^3*c^2*d^2*e^2-17*b^2*c^3*d^3*e+ 
8*b*c^4*d^4)/(4*a*c-b^2)^2/c^4)/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(-e^ 
4*(2*b*e-5*c*d)/c^3+1/3*(108*a^2*b*c^2*e^5-288*a^2*c^3*d*e^4-55*a*b^3*c*e^ 
5+180*a*b^2*c^2*d*e^4-132*a*b*c^3*d^2*e^3+128*a*c^4*d^3*e^2+7*b^5*e^5-25*b 
^4*c*d*e^4+17*b^3*c^2*d^2*e^3+16*b^2*c^3*d^3*e^2-64*b*c^4*d^4*e+32*c^5*d^5 
)/(4*a*c-b^2)^2/c^3-1/3/c^3*e*(36*a^2*b*c^2*e^4-108*a^2*c^3*d*e^3-11*a*b^3 
*c*e^4+27*a*b^2*c^2*d*e^3+24*a*b*c^3*d^2*e^2+4*a*c^4*d^3*e+b^5*e^4-4*b^4*c 
*d*e^3+6*b^3*c^2*d^2*e^2-17*b^2*c^3*d^3*e+8*b*c^4*d^4)/(4*a*c-b^2)^2+2/3/c 
^2*d*(36*a^2*c^2*e^4-33*a*b^2*c*e^4+60*a*b*c^2*d*e^3-60*a*c^3*d^2*e^2+5*b^ 
4*e^4-7*b^3*c*d*e^3-9*b^2*c^2*d^2*e^2+32*b*c^3*d^3*e-16*c^4*d^4)/(4*a*c-b^ 
2)^2)*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2336 vs. \(2 (740) = 1480\).

Time = 0.15 (sec) , antiderivative size = 2336, normalized size of antiderivative = 2.91 \[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(9/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

2/9*((16*a^2*c^5*d^5 - 40*a^2*b*c^4*d^4*e + 2*(11*a^2*b^2*c^3 + 36*a^3*c^4 
)*d^3*e^2 + (7*a^2*b^3*c^2 - 108*a^3*b*c^3)*d^2*e^3 + (11*a^2*b^4*c - 102* 
a^3*b^2*c^2 + 312*a^4*c^3)*d*e^4 - (8*a^2*b^5 - 69*a^3*b^3*c + 156*a^4*b*c 
^2)*e^5 + (16*c^7*d^5 - 40*b*c^6*d^4*e + 2*(11*b^2*c^5 + 36*a*c^6)*d^3*e^2 
 + (7*b^3*c^4 - 108*a*b*c^5)*d^2*e^3 + (11*b^4*c^3 - 102*a*b^2*c^4 + 312*a 
^2*c^5)*d*e^4 - (8*b^5*c^2 - 69*a*b^3*c^3 + 156*a^2*b*c^4)*e^5)*x^4 + 2*(1 
6*b*c^6*d^5 - 40*b^2*c^5*d^4*e + 2*(11*b^3*c^4 + 36*a*b*c^5)*d^3*e^2 + (7* 
b^4*c^3 - 108*a*b^2*c^4)*d^2*e^3 + (11*b^5*c^2 - 102*a*b^3*c^3 + 312*a^2*b 
*c^4)*d*e^4 - (8*b^6*c - 69*a*b^4*c^2 + 156*a^2*b^2*c^3)*e^5)*x^3 + (16*(b 
^2*c^5 + 2*a*c^6)*d^5 - 40*(b^3*c^4 + 2*a*b*c^5)*d^4*e + 2*(11*b^4*c^3 + 5 
8*a*b^2*c^4 + 72*a^2*c^5)*d^3*e^2 + (7*b^5*c^2 - 94*a*b^3*c^3 - 216*a^2*b* 
c^4)*d^2*e^3 + (11*b^6*c - 80*a*b^4*c^2 + 108*a^2*b^2*c^3 + 624*a^3*c^4)*d 
*e^4 - (8*b^7 - 53*a*b^5*c + 18*a^2*b^3*c^2 + 312*a^3*b*c^3)*e^5)*x^2 + 2* 
(16*a*b*c^5*d^5 - 40*a*b^2*c^4*d^4*e + 2*(11*a*b^3*c^3 + 36*a^2*b*c^4)*d^3 
*e^2 + (7*a*b^4*c^2 - 108*a^2*b^2*c^3)*d^2*e^3 + (11*a*b^5*c - 102*a^2*b^3 
*c^2 + 312*a^3*b*c^3)*d*e^4 - (8*a*b^6 - 69*a^2*b^4*c + 156*a^3*b^2*c^2)*e 
^5)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c 
)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d 
*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) 
+ 3*(16*a^2*c^5*d^4*e - 32*a^2*b*c^4*d^3*e^2 + 3*(3*a^2*b^2*c^3 + 20*a^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(9/2)/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(9/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(9/2)/(c*x^2 + b*x + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(9/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((e*x + d)^(9/2)/(c*x^2 + b*x + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int((d + e*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((d + e*x)^(9/2)/(a + b*x + c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (e x +d \right )^{\frac {9}{2}}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}d x \] Input:

int((e*x+d)^(9/2)/(c*x^2+b*x+a)^(5/2),x)
 

Output:

int((e*x+d)^(9/2)/(c*x^2+b*x+a)^(5/2),x)