3.4.99 \(\int \sqrt {d+e x} (b x+c x^2)^{5/2} \, dx\) [399]

3.4.99.1 Optimal result

Integrand size = 23, antiderivative size = 666 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^5 d^5-368 b c^4 d^4 e+303 b^2 c^3 d^3 e^2-22 b^3 c^2 d^2 e^3-17 b^4 c d e^4+24 b^5 e^5-3 c e \left (32 c^4 d^4-64 b c^3 d^3 e+21 b^2 c^2 d^2 e^2+11 b^3 c d e^3-24 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{9009 c^3 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^3 d^3-31 b c^2 d^2 e+9 b^2 c d e^2-18 b^3 e^3-14 c e \left (c^2 d^2-b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{9009 c^2 e^3}-\frac {10 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{143 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}}{13 e}-\frac {4 \sqrt {-b} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+79 b^2 c^2 d^2 e^2+49 b^3 c d e^3+24 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

output

2/13*(e*x+d)^(3/2)*(c*x^2+b*x)^(5/2)/e+10/9009*(16*c^3*d^3-31*b*c^2*d^2*e+ 
9*b^2*c*d*e^2-18*b^3*e^3-14*c*e*(3*b^2*e^2-b*c*d*e+c^2*d^2)*x)*(c*x^2+b*x) 
^(3/2)*(e*x+d)^(1/2)/c^2/e^3-10/143*(-b*e+2*c*d)*(c*x^2+b*x)^(5/2)*(e*x+d) 
^(1/2)/c/e-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+343*b^2*c^4*d^4*e^2-384*b*c^5*d^5*e+128*c^6*d^6)*EllipticE(c^(1/2 
)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*( 
e*x+d)^(1/2)/c^(7/2)/e^6/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/9009*d*(-b*e+ 
c*d)*(-b*e+2*c*d)*(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)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*( 
-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(7/2)/e^6/(e*x+d)^(1/2 
)/(c*x^2+b*x)^(1/2)+2/9009*(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*(-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)*x)*(e*x+d)^(1/2)*(c*x 
^2+b*x)^(1/2)/c^3/e^5
 

3.4.99.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.23 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (24 b^5 e^5-b^4 c e^4 (17 d+18 e x)+b^3 c^2 e^3 \left (-22 d^2+12 d e x+15 e^2 x^2\right )+b^2 c^3 e^2 \left (303 d^3-218 d^2 e x+178 d e^2 x^2+1113 e^3 x^3\right )+b c^4 e \left (-368 d^4+272 d^3 e x-225 d^2 e^2 x^2+196 d e^3 x^3+1701 e^4 x^4\right )+c^5 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )+\sqrt {\frac {b}{c}} \left (-2 \sqrt {\frac {b}{c}} \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) (b+c x) (d+e x)-2 i b e \left (128 c^6 d^6-384 b c^5 d^5 e+343 b^2 c^4 d^4 e^2-46 b^3 c^3 d^3 e^3-21 b^4 c^2 d^2 e^4-20 b^5 c d e^5+24 b^6 e^6\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (128 c^6 d^6-400 b c^5 d^5 e+383 b^2 c^4 d^4 e^2-70 b^3 c^3 d^3 e^3-25 b^4 c^2 d^2 e^4-64 b^5 c d e^5+48 b^6 e^6\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{9009 b c^3 e^6 x^3 (b+c x)^3 \sqrt {d+e x}} \]

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])
 

3.4.99.3 Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.05, 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)
 

3.4.99.3.1 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 120

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]
 

rule 122

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])
 

rule 126

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])
 

rule 127

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])
 

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 1169

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]
 

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 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]
 

3.4.99.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1727\) vs. \(2(600)=1200\).

Time = 1.96 (sec) , antiderivative size = 1728, normalized size of antiderivative = 2.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1728\)
elliptic	\(\text {Expression too large to display}\)	\(2334\)

input

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

output

2/9009*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(693*c^8*e^7*x^8+24*((c*x+b)/b)^(1/ 
2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2) 
,(b*e/(b*e-c*d))^(1/2))*b^7*c*d*e^6-23*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e- 
c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/ 
2))*b^6*c^2*d^2*e^5-20*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c* 
x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^3*d^3* 
e^4-395*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli 
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^4*d^4*e^3+1054*((c*x+ 
b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b) 
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^5*d^5*e^2-896*((c*x+b)/b)^(1/2)*(-c 
*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/ 
(b*e-c*d))^(1/2))*b^2*c^6*d^6*e-88*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d) 
)^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))* 
b^7*c*d*e^6-2*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2 
)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^6*c^2*d^2*e^5+2394* 
b*c^7*e^7*x^7+756*c^8*d*e^6*x^7+2814*b^2*c^6*e^7*x^6-7*c^8*d^2*e^5*x^6+112 
8*b^3*c^5*e^7*x^5+10*c^8*d^3*e^4*x^5-3*b^4*c^4*e^7*x^4-16*c^8*d^4*e^3*x^4+ 
6*b^5*c^3*e^7*x^3+32*c^8*d^5*e^2*x^3+24*b^6*c^2*e^7*x^2+128*c^8*d^6*e*x^2- 
50*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE 
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^3*d^3*e^4+778*((c*x+b)/...
 

3.4.99.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.23 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.12 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx=\frac {2 \, {\left ({\left (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}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (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}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (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 \, {\left (c^{7} d e^{6} + 27 \, b c^{6} e^{7}\right )} x^{4} - 7 \, {\left (10 \, c^{7} d^{2} e^{5} - 28 \, b c^{6} d e^{6} - 159 \, b^{2} c^{5} e^{7}\right )} x^{3} + {\left (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}\right )} x^{2} - 2 \, {\left (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}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{27027 \, c^{5} e^{7}} \]

input

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

3.4.99.6 Sympy [F]

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

input

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

output

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

3.4.99.7 Maxima [F]

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

input

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

output

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

3.4.99.8 Giac [F]

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

input

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

output

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

3.4.99.9 Mupad [F(-1)]

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)