Integrand size = 24, antiderivative size = 716 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(1/2)-10/63*(-14*c*e*x-15*b*e+16*c*d)*(c* x^2+b*x+a)^(3/2)*(e*x+d)^(1/2)/e^3-2/63*(128*c^3*d^3-b^3*e^3+3*b*c*e^2*(-3 6*a*e+37*b*d)-12*c^2*d*e*(-11*a*e+20*b*d)-3*c*e*(32*c^2*d^2+b^2*e^2-4*c*e* (-7*a*e+8*b*d))*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e^5+2/63*(128*c^4*d ^4-b^4*e^4-4*c^3*d^2*e*(-57*a*e+64*b*d)-b^2*c*e^3*(-15*a*e+7*b*d)+3*c^2*e^ 2*(28*a^2*e^2-76*a*b*d*e+45*b^2*d^2))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2) ^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d- e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2) *(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^2/e^6/(c*x^2+b*x+a)^(1/2)/(c*(e*x +d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/63*(-b*e+2*c*d)*(a*e^2-b*d*e +c*d^2)*(128*c^2*d^2-b^2*e^2-4*c*e*(-33*a*e+32*b*d))*EllipticF(1/2*((b+2*c *x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2 )^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2 )*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2 )^(1/2))))^(1/2)/c^2/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 34.22 (sec) , antiderivative size = 1140, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\frac {4 e^2 \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)+b^2 c e^3 (-7 b d+15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) (a+x (b+c x))}{\sqrt {d+e x}}-\frac {2 c e^2 (a+x (b+c x)) \left (63 c \left (c d^2+e (-b d+a e)\right )^2+\left (65 c^3 d^3-b^3 e^3+3 b c e^2 (16 b d-19 a e)+2 c^2 d e (-57 b d+43 a e)\right ) (d+e x)-c e \left (33 c^2 d^2-50 b c d e+15 b^2 e^2+28 a c e^2\right ) x (d+e x)+c^2 e^2 (17 c d-19 b e) x^2 (d+e x)-7 c^3 e^3 x^3 (d+e x)\right )}{\sqrt {d+e x}}-\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (-\left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (-128 c^4 d^4+b^4 e^4+4 c^3 d^2 e (64 b d-57 a e)+b^2 c e^3 (7 b d-15 a e)-3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )+\left (-b^5 e^5+b^4 e^4 \left (-6 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b^3 c e^3 \left (24 c d^2+16 a e^2+7 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )-b^2 c e^2 \left (16 c^2 d^3+135 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+15 a e^2 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (-12 a^2 c^2 e^5+64 c^3 d^3 e \sqrt {\left (b^2-4 a c\right ) e^2}+3 a c^2 d e^3 \left (-8 c d+19 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )-4 c^2 \left (32 c^2 d^4 \sqrt {\left (b^2-4 a c\right ) e^2}+3 a^2 e^4 \left (-8 c d+7 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a c d^2 e^2 \left (-16 c d+57 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{\sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{63 c^2 e^7 \sqrt {a+x (b+c x)}} \]
((4*e^2*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) + b^2*c*e^3 *(-7*b*d + 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*e^2))*(a + x*(b + c*x)))/Sqrt[d + e*x] - (2*c*e^2*(a + x*(b + c*x))*(63*c*(c*d^2 + e*(-(b*d) + a*e))^2 + (65*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(16*b*d - 19*a*e) + 2*c^2*d*e*(-57*b*d + 43*a*e))*(d + e*x) - c*e*(33*c^2*d^2 - 50*b*c*d*e + 15*b^2*e^2 + 28*a*c*e^2)*x*(d + e*x) + c^2*e^2*(17*c*d - 19*b*e)*x^2*(d + e*x) - 7*c^3*e^3*x^3*(d + e*x)))/Sqrt[d + e*x] - (I*(d + e*x)*Sqrt[1 - (2 *(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*(-((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(-1 28*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(64*b*d - 57*a*e) + b^2*c*e^3*(7*b*d - 15*a*e) - 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*e^2))*EllipticE[I*Ar cSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4* a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2 *c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]) + (-(b^5*e^5) + b^4*e^4*(-6*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b^3*c*e^3*(24*c*d^2 + 16*a*e^2 + 7*d*Sqrt[(b^2 - 4*a*c)*e^2]) - b^2*c*e^2*(16*c^2*d^3 + 135*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2 ] + 15*a*e^2*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(-12*a^2*c^2*e^5 + 64*c^3*d^3* e*Sqrt[(b^2 - 4*a*c)*e^2] + 3*a*c^2*d*e^3*(-8*c*d + 19*Sqrt[(b^2 - 4*a*c)* e^2])) - 4*c^2*(32*c^2*d^4*Sqrt[(b^2 - 4*a*c)*e^2] + 3*a^2*e^4*(-8*c*d ...
Time = 1.11 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1161, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{\sqrt {d+e x}}dx}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (-\frac {2 \int \frac {c \left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e-\left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {\int \frac {\left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e-\left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {2 \int \frac {\left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right )+5 c e (b d-2 a e) \left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e\right )+2 \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\int \frac {\left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right )+5 c e (b d-2 a e) \left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e\right )+2 \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\frac {2 \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {5 \left (-\frac {2 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (c x^2+b x+a\right )^{3/2}}{63 e^2}-\frac {\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-12 c^2 e (20 b d-11 a e) d-b^3 e^3+3 b c e^2 (37 b d-36 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d-b^2 e^2+132 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{21 e^2}\right )}{e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
(-2*(a + b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (5*((-2*Sqrt[d + e*x]*(16 *c*d - 15*b*e - 14*c*e*x)*(a + b*x + c*x^2)^(3/2))/(63*e^2) - ((2*Sqrt[d + e*x]*(128*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(37*b*d - 36*a*e) - 12*c^2*d*e*(2 0*b*d - 11*a*e) - 3*c*e*(32*c^2*d^2 + b^2*e^2 - 4*c*e*(8*b*d - 7*a*e))*x)* Sqrt[a + b*x + c*x^2])/(15*c*e^2) - ((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*c^4 *d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e ) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*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 + 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]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a *c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - 128*b*c*d*e - b^2 *e^2 + 132*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[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*e^2)))/e
3.25.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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 + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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]
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])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2071\) vs. \(2(648)=1296\).
Time = 3.33 (sec) , antiderivative size = 2072, normalized size of antiderivative = 2.89
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2072\) |
| elliptic | \(\text {Expression too large to display}\) | \(2528\) |
| default | \(\text {Expression too large to display}\) | \(9187\) |
2/63/c*(7*c^3*e^3*x^3+19*b*c^2*e^3*x^2-17*c^3*d*e^2*x^2+28*a*c^2*e^3*x+15* b^2*c*e^3*x-50*b*c^2*d*e^2*x+33*c^3*d^2*e*x+57*a*b*c*e^3-86*a*c^2*d*e^2+b^ 3*e^3-48*b^2*c*d*e^2+114*b*c^2*d^2*e-65*c^3*d^3)*(e*x+d)^(1/2)*(c*x^2+b*x+ a)^(1/2)/e^5+1/63/c/e^5*(2*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e ^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^ 3*d^3*e+193*c^4*d^4)*(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^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e- 1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(- d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b* d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d /e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/ c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/ 2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/ 2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))+ 63*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4-6*a*b*c*d^3*e^3+ 3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e^2-3*b*c^2*d^5*e+c^3*d^6)*c/e*(-2 *(c*e*x^2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/ 2)+2*(-(b*e-c*d)/(a*e^2-b*d*e+c*d^2)+b*e/(a*e^2-b*d*e+c*d^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^2)^(1/2))/c))^(1/2)* ((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.29 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \]
-2/189*((256*c^5*d^6 - 640*b*c^4*d^5*e + 2*(239*b^2*c^3 + 324*a*c^4)*d^4*e ^2 - (77*b^3*c^2 + 972*a*b*c^3)*d^3*e^3 - (13*b^4*c - 258*a*b^2*c^2 - 456* a^2*c^3)*d^2*e^4 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d*e^5 + (256*c^5*d ^5*e - 640*b*c^4*d^4*e^2 + 2*(239*b^2*c^3 + 324*a*c^4)*d^3*e^3 - (77*b^3*c ^2 + 972*a*b*c^3)*d^2*e^4 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d*e^5 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*e^6)*x)*sqrt(c*e)*weierstrassPInve rse(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)) + 6*(128*c^5*d^5*e - 256*b*c^4*d^ 4*e^2 + 3*(45*b^2*c^3 + 76*a*c^4)*d^3*e^3 - (7*b^3*c^2 + 228*a*b*c^3)*d^2* e^4 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d*e^5 + (128*c^5*d^4*e^2 - 256*b *c^4*d^3*e^3 + 3*(45*b^2*c^3 + 76*a*c^4)*d^2*e^4 - (7*b^3*c^2 + 228*a*b*c^ 3)*d*e^5 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*e^6)*x)*sqrt(c*e)*weierstra ssZeta(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*( 7*c^5*e^6*x^4 - 128*c^5*d^4*e^2 + 240*b*c^4*d^3*e^3 - 63*a^2*c^3*e^6 - (11 1*b^2*c^3 + 212*a*c^4)*d^2*e^4 + (b^3*c^2 + 183*a*b*c^3)*d*e^5 - (10*c^...
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]