Integrand size = 23, antiderivative size = 817 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=-\frac {4 \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) x \sqrt {d+e x}}{9009 c^3 e^6 \sqrt {b x+c x^2}}+\frac {2 \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{9009 c^3 e^5}-\frac {4 \left (48 c^4 d^4-136 b c^3 d^3 e+109 b^2 c^2 d^2 e^2-6 b^3 c d e^3+9 b^4 e^4\right ) x \sqrt {d+e x} \sqrt {b x+c x^2}}{9009 c^2 e^4}+\frac {2 \left (80 c^3 d^3-225 b c^2 d^2 e+178 b^2 c d e^2+15 b^3 e^3\right ) x^2 \sqrt {d+e x} \sqrt {b x+c x^2}}{9009 c e^3}-\frac {2 (2 c d-5 b e) (5 c d+3 b e) x^3 \sqrt {d+e x} \sqrt {b x+c x^2}}{1287 e^2}+\frac {2 (c d+5 b e) x^2 \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{143 e}+\frac {2}{13} x \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}+\frac {4 \sqrt {b} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5\right ) \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{9009 c^{7/2} e^5 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:
-4/9009*(24*b^6*e^6-20*b^5*c*d*e^5-21*b^4*c^2*d^2*e^4-46*b^3*c^3*d^3*e^3+3 43*b^2*c^4*d^4*e^2-384*b*c^5*d^5*e+128*c^6*d^6)*x*(e*x+d)^(1/2)/c^3/e^6/(c *x^2+b*x)^(1/2)+2/9009*(24*b^5*e^5-17*b^4*c*d*e^4-22*b^3*c^2*d^2*e^3+303*b ^2*c^3*d^3*e^2-368*b*c^4*d^4*e+128*c^5*d^5)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2 )/c^3/e^5-4/9009*(9*b^4*e^4-6*b^3*c*d*e^3+109*b^2*c^2*d^2*e^2-136*b*c^3*d^ 3*e+48*c^4*d^4)*x*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^2/e^4+2/9009*(15*b^3*e ^3+178*b^2*c*d*e^2-225*b*c^2*d^2*e+80*c^3*d^3)*x^2*(e*x+d)^(1/2)*(c*x^2+b* x)^(1/2)/c/e^3-2/1287*(-5*b*e+2*c*d)*(3*b*e+5*c*d)*x^3*(e*x+d)^(1/2)*(c*x^ 2+b*x)^(1/2)/e^2+2/143*(5*b*e+c*d)*x^2*(e*x+d)^(1/2)*(c*x^2+b*x)^(3/2)/e+2 /13*x*(e*x+d)^(1/2)*(c*x^2+b*x)^(5/2)+4/9009*b^(1/2)*(24*b^6*e^6-20*b^5*c* d*e^5-21*b^4*c^2*d^2*e^4-46*b^3*c^3*d^3*e^3+343*b^2*c^4*d^4*e^2-384*b*c^5* d^5*e+128*c^6*d^6)*x^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2)*x^(1/2)/b^(1/2) /(1+c*x/b)^(1/2),(1-b*e/c/d)^(1/2))/c^(7/2)/e^6/(b*(e*x+d)/d/(c*x+b))^(1/2 )/(c*x^2+b*x)^(1/2)-2/9009*b^(3/2)*(24*b^5*e^5-17*b^4*c*d*e^4-22*b^3*c^2*d ^2*e^3+303*b^2*c^3*d^3*e^2-368*b*c^4*d^4*e+128*c^5*d^5)*x^(1/2)*(e*x+d)^(1 /2)*InverseJacobiAM(arctan(c^(1/2)*x^(1/2)/b^(1/2)),(1-b*e/c/d)^(1/2))/c^( 7/2)/e^5/(b*(e*x+d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 20.73 (sec) , antiderivative size = 663, normalized size of antiderivative = 0.81 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (24 b^5 e^5-b^4 c e^4 (17 d+18 e x)+b^3 c^2 e^3 \left (-22 d^2+12 d e x+15 e^2 x^2\right )+b^2 c^3 e^2 \left (303 d^3-218 d^2 e x+178 d e^2 x^2+1113 e^3 x^3\right )+b c^4 e \left (-368 d^4+272 d^3 e x-225 d^2 e^2 x^2+196 d e^3 x^3+1701 e^4 x^4\right )+c^5 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )+\sqrt {\frac {b}{c}} \left (-2 \sqrt {\frac {b}{c}} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) (b+c x) (d+e x)-2 i b e \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (128 c^6 d^6-400 b c^5 d^5 e+383 b^2 c^4 d^4 e^2-70 b^3 c^3 d^3 e^3-25 b^4 c^2 d^2 e^4-64 b^5 c d e^5+48 b^6 e^6\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{9009 b c^3 e^6 x^3 (b+c x)^3 \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^(5/2),x]
Output:
(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(24*b^5*e^5 - b^4*c*e^4* (17*d + 18*e*x) + b^3*c^2*e^3*(-22*d^2 + 12*d*e*x + 15*e^2*x^2) + b^2*c^3* e^2*(303*d^3 - 218*d^2*e*x + 178*d*e^2*x^2 + 1113*e^3*x^3) + b*c^4*e*(-368 *d^4 + 272*d^3*e*x - 225*d^2*e^2*x^2 + 196*d*e^3*x^3 + 1701*e^4*x^4) + c^5 *(128*d^5 - 96*d^4*e*x + 80*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(128*c^6*d^6 - 384*b*c^5*d^5*e + 3 43*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d* e^5 + 24*b^6*e^6)*(b + c*x)*(d + e*x) - (2*I)*b*e*(128*c^6*d^6 - 384*b*c^5 *d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 2 0*b^5*c*d*e^5 + 24*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*El lipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(128*c^6*d^6 - 400*b*c^5*d^5*e + 383*b^2*c^4*d^4*e^2 - 70*b^3*c^3*d^3*e^3 - 25*b^4*c^2*d^ 2*e^4 - 64*b^5*c*d*e^5 + 48*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x ^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(9009*b*c^3 *e^6*x^3*(b + c*x)^3*Sqrt[d + e*x])
Time = 1.75 (sec) , antiderivative size = 697, normalized size of antiderivative = 0.85, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {1162, 1236, 27, 1231, 27, 1231, 27, 1269, 1169, 122, 120, 127, 126}
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.
| \(\displaystyle \int \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \int \sqrt {d+e x} (b d+(2 c d-b e) x) \left (c x^2+b x\right )^{3/2}dx}{13 e}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {2 \int \frac {\left (b d (c d+5 b e)+2 \left (c^2 d^2-b c e d+3 b^2 e^2\right ) x\right ) \left (c x^2+b x\right )^{3/2}}{2 \sqrt {d+e x}}dx}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\int \frac {\left (b d (c d+5 b e)+2 \left (c^2 d^2-b c e d+3 b^2 e^2\right ) x\right ) \left (c x^2+b x\right )^{3/2}}{\sqrt {d+e x}}dx}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {-\frac {2 \int -\frac {\left (b d \left (16 c^3 d^3-31 b c^2 e d^2+9 b^2 c e^2 d-18 b^3 e^3\right )+\left (32 c^4 d^4-64 b c^3 e d^3+21 b^2 c^2 e^2 d^2+11 b^3 c e^3 d-24 b^4 e^4\right ) x\right ) \sqrt {c x^2+b x}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\int \frac {\left (b d \left (16 c^3 d^3-31 b c^2 e d^2+9 b^2 c e^2 d-18 b^3 e^3\right )+\left (32 c^4 d^4-64 b c^3 e d^3+21 b^2 c^2 e^2 d^2+11 b^3 c e^3 d-24 b^4 e^4\right ) x\right ) \sqrt {c x^2+b x}}{\sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {-\frac {2 \int -\frac {b d \left (128 c^5 d^5-368 b c^4 e d^4+303 b^2 c^3 e^2 d^3-22 b^3 c^2 e^3 d^2-17 b^4 c e^4 d+24 b^5 e^5\right )+2 \left (128 c^6 d^6-384 b c^5 e d^5+343 b^2 c^4 e^2 d^4-46 b^3 c^3 e^3 d^3-21 b^4 c^2 e^4 d^2-20 b^5 c e^5 d+24 b^6 e^6\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\int \frac {b d \left (128 c^5 d^5-368 b c^4 e d^4+303 b^2 c^3 e^2 d^3-22 b^3 c^2 e^3 d^2-17 b^4 c e^4 d+24 b^5 e^5\right )+2 \left (128 c^6 d^6-384 b c^5 e d^5+343 b^2 c^4 e^2 d^4-46 b^3 c^3 e^3 d^3-21 b^4 c^2 e^4 d^2-20 b^5 c e^5 d+24 b^6 e^6\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\frac {2 \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\frac {2 \sqrt {x} \sqrt {b+c x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {5 \left (\frac {\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{15 c e^2}}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{63 c e^2}}{11 c}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x} (2 c d-b e)}{11 c}\right )}{13 e}\) |
Input:
Int[Sqrt[d + e*x]*(b*x + c*x^2)^(5/2),x]
Output:
(2*(d + e*x)^(3/2)*(b*x + c*x^2)^(5/2))/(13*e) - (5*((2*(2*c*d - b*e)*Sqrt [d + e*x]*(b*x + c*x^2)^(5/2))/(11*c) + ((-2*Sqrt[d + e*x]*(16*c^3*d^3 - 3 1*b*c^2*d^2*e + 9*b^2*c*d*e^2 - 18*b^3*e^3 - 14*c*e*(c^2*d^2 - b*c*d*e + 3 *b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(63*c*e^2) + ((-2*Sqrt[d + e*x]*(128*c^5 *d^5 - 368*b*c^4*d^4*e + 303*b^2*c^3*d^3*e^2 - 22*b^3*c^2*d^2*e^3 - 17*b^4 *c*d*e^4 + 24*b^5*e^5 - 3*c*e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 21*b^2*c^2*d^ 2*e^2 + 11*b^3*c*d*e^3 - 24*b^4*e^4)*x)*Sqrt[b*x + c*x^2])/(15*c*e^2) + (( 4*Sqrt[-b]*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c ^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*Sqrt[x]*Sqr t[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[- b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 79*b^2*c^2 *d^2*e^2 + 49*b^3*c*d*e^3 + 24*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqr t[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(15*c*e^2))/(21*c*e^2))/(11*c)))/ (13*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 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 + 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]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[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*(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])
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. \(1727\) vs. \(2(748)=1496\).
Time = 1.96 (sec) , antiderivative size = 1728, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1728\) |
| elliptic | \(\text {Expression too large to display}\) | \(2334\) |
Input:
int((e*x+d)^(1/2)*(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9009*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(48*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b *e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d)) ^(1/2))*b^7*d*e^7-48*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d )^(1/2)*EllipticE(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b^7*d*e^7+756* c^7*d*e^7*x^7-36*b*c^6*d^2*e^6*x^5+1318*b^3*c^4*d*e^7*x^4-69*b^2*c^5*d^2*e ^6*x^4+57*b*c^6*d^3*e^5*x^4-8*b^4*c^3*d*e^7*x^3-50*b^3*c^4*d^2*e^6*x^3+132 *b^2*c^5*d^3*e^5*x^3-112*b*c^6*d^4*e^4*x^3-11*b^5*c^2*d*e^7*x^2-27*b^4*c^3 *d^2*e^6*x^2+63*b^3*c^4*d^3*e^5*x^2+207*b^2*c^5*d^4*e^4*x^2-336*b*c^6*d^5* e^3*x^2+24*b^6*c*d*e^7*x-17*b^5*c^2*d^2*e^6*x-22*b^4*c^3*d^3*e^5*x+303*b^3 *c^4*d^4*e^4*x-368*b^2*c^5*d^5*e^3*x+128*b*c^6*d^6*e^2*x-3*b^4*c^3*e^8*x^4 -16*c^7*d^4*e^4*x^4+6*b^5*c^2*e^8*x^3+32*c^7*d^5*e^3*x^3+24*b^6*c*e^8*x^2+ 128*c^7*d^6*e^2*x^2+2394*b*c^6*e^8*x^7+2814*b^2*c^5*e^8*x^6-7*c^7*d^2*e^6* x^6+1128*b^3*c^4*e^8*x^5+10*c^7*d^3*e^5*x^5+693*c^7*e^8*x^8+256*((e*x+d)/d )^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticE(((e*x+d)/d)^( 1/2),(-d*c/(b*e-c*d))^(1/2))*c^7*d^8-64*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e- c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1 /2))*b^6*c*d^2*e^6-25*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/ d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b^5*c^2*d^3*e ^5-70*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*Ellipti cF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b^4*c^3*d^4*e^4+383*((e*x+...
Time = 0.23 (sec) , antiderivative size = 747, normalized size of antiderivative = 0.91 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(c*x^2+b*x)^(5/2),x, algorithm="fricas")
Output:
2/27027*((256*c^7*d^7 - 896*b*c^6*d^6*e + 1022*b^2*c^5*d^5*e^2 - 315*b^3*c ^4*d^4*e^3 - 68*b^4*c^3*d^3*e^4 - 31*b^5*c^2*d^2*e^5 - 64*b^6*c*d*e^6 + 48 *b^7*e^7)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/ (c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/( c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 6*(128*c^7*d^6*e - 384*b*c^6* d^5*e^2 + 343*b^2*c^5*d^4*e^3 - 46*b^3*c^4*d^3*e^4 - 21*b^4*c^3*d^2*e^5 - 20*b^5*c^2*d*e^6 + 24*b^6*c*e^7)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d *e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b ^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(693*c^7*e^7*x^5 + 128*c^7*d^5*e^2 - 368*b*c^6*d^4*e^3 + 303*b^2*c^5*d^3*e^4 - 22*b^3*c^4*d ^2*e^5 - 17*b^4*c^3*d*e^6 + 24*b^5*c^2*e^7 + 63*(c^7*d*e^6 + 27*b*c^6*e^7) *x^4 - 7*(10*c^7*d^2*e^5 - 28*b*c^6*d*e^6 - 159*b^2*c^5*e^7)*x^3 + (80*c^7 *d^3*e^4 - 225*b*c^6*d^2*e^5 + 178*b^2*c^5*d*e^6 + 15*b^3*c^4*e^7)*x^2 - 2 *(48*c^7*d^4*e^3 - 136*b*c^6*d^3*e^4 + 109*b^2*c^5*d^2*e^5 - 6*b^3*c^4*d*e ^6 + 9*b^4*c^3*e^7)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^5*e^7)
\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}\, dx \] Input:
integrate((e*x+d)**(1/2)*(c*x**2+b*x)**(5/2),x)
Output:
Integral((x*(b + c*x))**(5/2)*sqrt(d + e*x), x)
\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d} \,d x } \] Input:
integrate((e*x+d)^(1/2)*(c*x^2+b*x)^(5/2),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x)^(5/2)*sqrt(e*x + d), x)
\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d} \,d x } \] Input:
integrate((e*x+d)^(1/2)*(c*x^2+b*x)^(5/2),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x)^(5/2)*sqrt(e*x + d), x)
Timed out. \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{5/2}\,\sqrt {d+e\,x} \,d x \] Input:
int((b*x + c*x^2)^(5/2)*(d + e*x)^(1/2),x)
Output:
int((b*x + c*x^2)^(5/2)*(d + e*x)^(1/2), x)
\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\text {too large to display} \] Input:
int((e*x+d)^(1/2)*(c*x^2+b*x)^(5/2),x)
Output:
(54*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**5*d*e**4 - 36*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**5*e**5*x - 36*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b* *4*c*d**2*e**3 - 12*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**4*c*d*e**4*x + 30*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**4*c*e**5*x**2 + 654*sqrt(x)*sqrt (d + e*x)*sqrt(b + c*x)*b**3*c**2*d**3*e**2 - 412*sqrt(x)*sqrt(d + e*x)*sq rt(b + c*x)*b**3*c**2*d**2*e**3*x + 386*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x )*b**3*c**2*d*e**4*x**2 + 2226*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**3*c* *2*e**5*x**3 - 816*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c**3*d**4*e + 108*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c**3*d**3*e**2*x - 94*sqrt(x) *sqrt(d + e*x)*sqrt(b + c*x)*b**2*c**3*d**2*e**3*x**2 + 2618*sqrt(x)*sqrt( d + e*x)*sqrt(b + c*x)*b**2*c**3*d*e**4*x**3 + 3402*sqrt(x)*sqrt(d + e*x)* sqrt(b + c*x)*b**2*c**3*e**5*x**4 + 288*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x )*b*c**4*d**5 + 352*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**4*d**4*e*x - 290*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**4*d**3*e**2*x**2 + 252*sqrt(x )*sqrt(d + e*x)*sqrt(b + c*x)*b*c**4*d**2*e**3*x**3 + 3528*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**4*d*e**4*x**4 + 1386*sqrt(x)*sqrt(d + e*x)*sqrt( b + c*x)*b*c**4*e**5*x**5 - 192*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**5*d **5*x + 160*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**5*d**4*e*x**2 - 140*sqr t(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**5*d**3*e**2*x**3 + 126*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**5*d**2*e**3*x**4 + 1386*sqrt(x)*sqrt(d + e*x)*s...