\(\int \frac {(a+b x+c x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [648]

Optimal result

Integrand size = 24, antiderivative size = 827 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^4 d^4+2 b^4 e^4-4 c^3 d^2 e (64 b d-69 a e)+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (41 b^2 d^2-92 a b d e+60 a^2 e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/693*(e*x+d)^(1/2)*(128*c^4*d^4-4*b^4*e^4-4*c^3*d^2*e*(-69*a*e+76*b*d)-b^ 
2*c*e^3*(-27*a*e+7*b*d)+3*c^2*e^2*(60*a^2*e^2-124*a*b*d*e+65*b^2*d^2)-12*c 
*e*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))*x)*(c*x^2+b*x+a)^(1 
/2)/c^2/e^5+10/693*(e*x+d)^(1/2)*(16*c^2*d^2+3*b^2*e^2-c*e*(-18*a*e+23*b*d 
)-7*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/c/e^3+2/11*(e*x+d)^(1/2)*(c*x^ 
2+b*x+a)^(5/2)/e-1/693*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(128*c^4*d^ 
4+8*b^4*e^4+b^2*c*e^3*(-93*a*e+29*b*d)-4*c^3*d^2*e*(-93*a*e+64*b*d)+3*c^2* 
e^2*(124*a^2*e^2-124*a*b*d*e+33*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/e^6/(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)+4/693*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(128*c^4*d^4+2*b^ 
4*e^4-4*c^3*d^2*e*(-69*a*e+64*b*d)+b^2*c*e^3*(-21*a*e+5*b*d)+3*c^2*e^2*(60 
*a^2*e^2-92*a*b*d*e+41*b^2*d^2))*(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/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 35.42 (sec) , antiderivative size = 10879, normalized size of antiderivative = 13.15 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1162, 1231, 27, 1231, 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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {2 \int \frac {\left (9 c e (b d-2 a e)^2-2 (2 c d-b e) \left (b d \left (4 c d-\frac {3 b e}{2}\right )-\frac {1}{2} a e (2 c d+b e)\right )-4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (18 a c e^2+3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{63 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {\int \frac {\left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (-3 d e b^2+8 c d^2 b-a e^2 b-2 a c d e\right )-4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (18 a c e^2+3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{63 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c^2 e^2 \left (60 a^2 e^2-124 a b d e+65 b^2 d^2\right )-12 c e x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-b^2 c e^3 (7 b d-27 a e)-4 c^3 d^2 e (76 b d-69 a e)-4 b^4 e^4+128 c^4 d^4\right )}{15 c e^2}-\frac {2 \int \frac {4 (2 c d-b e) \left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (-3 d e b^2+8 c d^2 b-a e^2 b-2 a c d e\right )\right )+(2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (18 a c e^2+3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{63 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c^2 e^2 \left (60 a^2 e^2-124 a b d e+65 b^2 d^2\right )-12 c e x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-b^2 c e^3 (7 b d-27 a e)-4 c^3 d^2 e (76 b d-69 a e)-4 b^4 e^4+128 c^4 d^4\right )}{15 c e^2}-\frac {\int \frac {4 (2 c d-b e) \left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (-3 d e b^2+8 c d^2 b-a e^2 b-2 a c d e\right )\right )+(2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (18 a c e^2+3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{63 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c^2 e^2 \left (60 a^2 e^2-124 a b d e+65 b^2 d^2\right )-12 c e x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-b^2 c e^3 (7 b d-27 a e)-4 c^3 d^2 e (76 b d-69 a e)-4 b^4 e^4+128 c^4 d^4\right )}{15 c e^2}-\frac {\frac {(2 c d-b e) \left (3 c^2 e^2 \left (124 a^2 e^2-124 a b d e+33 b^2 d^2\right )+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+8 b^4 e^4+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (a e^2-b d e+c d^2\right ) \left (180 a^2 c^2 e^4-21 a b^2 c e^4-276 a b c^2 d e^3+276 a c^3 d^2 e^2+2 b^4 e^4+5 b^3 c d e^3+123 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{15 c e^2}}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (18 a c e^2+3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{63 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {2 \sqrt {d+e x} \left (16 c^2 d^2-23 b c e d+3 b^2 e^2+18 a c e^2-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 c e^2}-\frac {\frac {2 \sqrt {d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-256 b c^3 e d^3+276 a c^3 e^2 d^2+123 b^2 c^2 e^2 d^2-276 a b c^2 e^3 d+5 b^3 c e^3 d+2 b^4 e^4+180 a^2 c^2 e^4-21 a b^2 c e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{21 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {2 \sqrt {d+e x} \left (16 c^2 d^2-23 b c e d+3 b^2 e^2+18 a c e^2-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 c e^2}-\frac {\frac {2 \sqrt {d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-256 b c^3 e d^3+276 a c^3 e^2 d^2+123 b^2 c^2 e^2 d^2-276 a b c^2 e^3 d+5 b^3 c e^3 d+2 b^4 e^4+180 a^2 c^2 e^4-21 a b^2 c e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{21 c e^2}\right )}{11 e}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}-\frac {5 \left (-\frac {2 \sqrt {d+e x} \left (16 c^2 d^2-23 b c e d+3 b^2 e^2+18 a c e^2-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 c e^2}-\frac {\frac {2 \sqrt {d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-256 b c^3 e d^3+276 a c^3 e^2 d^2+123 b^2 c^2 e^2 d^2-276 a b c^2 e^3 d+5 b^3 c e^3 d+2 b^4 e^4+180 a^2 c^2 e^4-21 a b^2 c e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{21 c e^2}\right )}{11 e}\)

Input:

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

Output:

(2*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*e) - (5*((-2*Sqrt[d + e*x]*( 
16*c^2*d^2 - 23*b*c*d*e + 3*b^2*e^2 + 18*a*c*e^2 - 7*c*e*(2*c*d - b*e)*x)* 
(a + b*x + c*x^2)^(3/2))/(63*c*e^2) - ((2*Sqrt[d + e*x]*(128*c^4*d^4 - 4*b 
^4*e^4 - 4*c^3*d^2*e*(76*b*d - 69*a*e) - b^2*c*e^3*(7*b*d - 27*a*e) + 3*c^ 
2*e^2*(65*b^2*d^2 - 124*a*b*d*e + 60*a^2*e^2) - 12*c*e*(2*c*d - b*e)*(4*c^ 
2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*x)*Sqrt[a + b*x + c*x^2])/(15*c*e^2 
) - ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^4*d^4 + 8*b^4*e^4 + b 
^2*c*e^3*(29*b*d - 93*a*e) - 4*c^3*d^2*e*(64*b*d - 93*a*e) + 3*c^2*e^2*(33 
*b^2*d^2 - 124*a*b*d*e + 124*a^2*e^2))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + 
c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(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 + Sq 
rt[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]) - (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b* 
d*e + a*e^2)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 123*b^2*c^2*d^2*e^2 + 276*a* 
c^3*d^2*e^2 + 5*b^3*c*d*e^3 - 276*a*b*c^2*d*e^3 + 2*b^4*e^4 - 21*a*b^2*c*e 
^4 + 180*a^2*c^2*e^4)*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[ArcSin[Sqrt[(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[d + e*x]*Sqrt[a + 
 b*x + c*x^2]))/(15*c*e^2))/(21*c*e^2)))/(11*e)
 

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 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2543\) vs. \(2(761)=1522\).

Time = 8.99 (sec) , antiderivative size = 2544, normalized size of antiderivative = 3.08

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2544\)
risch	\(\text {Expression too large to display}\)	\(4982\)
default	\(\text {Expression too large to display}\)	\(12153\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),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)*(2/11*c^2/ 
e*x^4*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/9*(3*b*c^2-2/11/e* 
c^2*(5*b*e+5*c*d))/c/e*x^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2) 
+2/7*(3*a*c^2+3*b^2*c-2/11*c^2/e*(9/2*a*e+9/2*b*d)-2/9*(3*b*c^2-2/11/e*c^2 
*(5*b*e+5*c*d))/c/e*(4*b*e+4*c*d))/c/e*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+ 
b*d*x+a*d)^(1/2)+2/5*(6*a*b*c+b^3-8/11*c^2/e*a*d-2/9*(3*b*c^2-2/11/e*c^2*( 
5*b*e+5*c*d))/c/e*(7/2*a*e+7/2*b*d)-2/7*(3*a*c^2+3*b^2*c-2/11*c^2/e*(9/2*a 
*e+9/2*b*d)-2/9*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c*d))/c/e*(4*b*e+4*c*d))/c/e* 
(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*( 
3*a^2*c+3*a*b^2-2/3*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c*d))/c/e*a*d-2/7*(3*a*c^ 
2+3*b^2*c-2/11*c^2/e*(9/2*a*e+9/2*b*d)-2/9*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c* 
d))/c/e*(4*b*e+4*c*d))/c/e*(5/2*a*e+5/2*b*d)-2/5*(6*a*b*c+b^3-8/11*c^2/e*a 
*d-2/9*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c*d))/c/e*(7/2*a*e+7/2*b*d)-2/7*(3*a*c 
^2+3*b^2*c-2/11*c^2/e*(9/2*a*e+9/2*b*d)-2/9*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c 
*d))/c/e*(4*b*e+4*c*d))/c/e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3 
+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(a^3-2/5*(6*a*b*c+b^3-8/11*c^2/e 
*a*d-2/9*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c*d))/c/e*(7/2*a*e+7/2*b*d)-2/7*(3*a 
*c^2+3*b^2*c-2/11*c^2/e*(9/2*a*e+9/2*b*d)-2/9*(3*b*c^2-2/11/e*c^2*(5*b*e+5 
*c*d))/c/e*(4*b*e+4*c*d))/c/e*(3*b*e+3*c*d))/c/e*a*d-2/3*(3*a^2*c+3*a*b^2- 
2/3*(3*b*c^2-2/11/e*c^2*(5*b*e+5*c*d))/c/e*a*d-2/7*(3*a*c^2+3*b^2*c-2/1...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 914, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/2079*((256*c^6*d^6 - 768*b*c^5*d^5*e + 6*(121*b^2*c^4 + 156*a*c^5)*d^4*e 
^2 - 4*(43*b^3*c^3 + 468*a*b*c^4)*d^3*e^3 - 3*(11*b^4*c^2 - 260*a*b^2*c^3 
- 416*a^2*c^4)*d^2*e^4 - 3*(3*b^5*c - 52*a*b^3*c^2 + 416*a^2*b*c^3)*d*e^5 
- (8*b^6 - 105*a*b^4*c + 498*a^2*b^2*c^2 - 1080*a^3*c^3)*e^6)*sqrt(c*e)*we 
ierstrassPInverse(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*(256*c^6*d^5*e 
- 640*b*c^5*d^4*e^2 + 2*(227*b^2*c^4 + 372*a*c^5)*d^3*e^3 - (41*b^3*c^3 + 
1116*a*b*c^4)*d^2*e^4 - (13*b^4*c^2 - 186*a*b^2*c^3 - 744*a^2*c^4)*d*e^5 - 
 (8*b^5*c - 93*a*b^3*c^2 + 372*a^2*b*c^3)*e^6)*sqrt(c*e)*weierstrassZeta(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 
), 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*(63*c^6*e 
^6*x^4 + 128*c^6*d^4*e^2 - 304*b*c^5*d^3*e^3 + (195*b^2*c^4 + 356*a*c^5)*d 
^2*e^4 - (7*b^3*c^3 + 487*a*b*c^4)*d*e^5 - (4*b^4*c^2 - 42*a*b^2*c^3 - 333 
*a^2*c^4)*e^6 - 7*(10*c^6*d*e^5 - 23*b*c^5*e^6)*x^3 + (80*c^6*d^2*e^4 - 18 
5*b*c^5*d*e^5 + (113*b^2*c^4 + 216*a*c^5)*e^6)*x^2 - (96*c^6*d^3*e^3 - 224 
*b*c^5*d^2*e^4 + (139*b^2*c^4 + 262*a*c^5)*d*e^5 - (3*b^3*c^3 + 347*a*b...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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