\(\int \frac {(b x+c x^2)^{5/2}}{d+e x} \, dx\) [152]

Optimal result

Integrand size = 21, antiderivative size = 380 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right ) \sqrt {b x+c x^2}}{128 c^2 e^5}-\frac {\left (96 c^3 d^3-208 b c^2 d^2 e+118 b^2 c d e^2-3 b^3 e^3\right ) x \sqrt {b x+c x^2}}{192 c e^4}+\frac {\left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x^2 \sqrt {b x+c x^2}}{48 e^3}-\frac {(2 c d-b e) x \left (b x+c x^2\right )^{3/2}}{8 e^2}+\frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {(2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+3 b^4 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{5/2} e^6}+\frac {2 d^{5/2} (c d-b e)^{5/2} \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{e^6} \] Output:

1/128*(-3*b^4*e^4-10*b^3*c*d*e^3+176*b^2*c^2*d^2*e^2-288*b*c^3*d^3*e+128*c 
^4*d^4)*(c*x^2+b*x)^(1/2)/c^2/e^5-1/192*(-3*b^3*e^3+118*b^2*c*d*e^2-208*b* 
c^2*d^2*e+96*c^3*d^3)*x*(c*x^2+b*x)^(1/2)/c/e^4+1/48*(3*b^2*e^2-22*b*c*d*e 
+16*c^2*d^2)*x^2*(c*x^2+b*x)^(1/2)/e^3-1/8*(-b*e+2*c*d)*x*(c*x^2+b*x)^(3/2 
)/e^2+1/5*(c*x^2+b*x)^(5/2)/e-1/128*(-b*e+2*c*d)*(3*b^4*e^4+16*b^3*c*d*e^3 
+112*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*arctanh(c^(1/2)*x/(c*x^2 
+b*x)^(1/2))/c^(5/2)/e^6+2*d^(5/2)*(-b*e+c*d)^(5/2)*arctanh((-b*e+c*d)^(1/ 
2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/e^6
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.96 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.64 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {(x (b+c x))^{5/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (-45 b^4 e^4+30 b^3 c e^3 (-5 d+e x)+4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (-270 d^3+130 d^2 e x-85 d e^2 x^2+63 e^3 x^3\right )+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-3840 c^{3/2} d^{3/2} (c d-b e)^2 \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )-3840 c^{3/2} d^{3/2} (c d-b e)^2 \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+30 \left (256 c^5 d^5-640 b c^4 d^4 e+480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )\right )}{1920 c^{5/2} e^6 x^{5/2} (b+c x)^{5/2}} \] Input:

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

Output:

((x*(b + c*x))^(5/2)*(Sqrt[c]*e*Sqrt[x]*Sqrt[b + c*x]*(-45*b^4*e^4 + 30*b^ 
3*c*e^3*(-5*d + e*x) + 4*b^2*c^2*e^2*(660*d^2 - 295*d*e*x + 186*e^2*x^2) + 
 16*b*c^3*e*(-270*d^3 + 130*d^2*e*x - 85*d*e^2*x^2 + 63*e^3*x^3) + 32*c^4* 
(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) - 3840 
*c^(3/2)*d^(3/2)*(c*d - b*e)^2*(c*d - b*e - I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b 
*e])*Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[( 
Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqr 
t[d]*(-Sqrt[b] + Sqrt[b + c*x]))] - 3840*c^(3/2)*d^(3/2)*(c*d - b*e)^2*(c* 
d - b*e + I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b*e + (2*I)*S 
qrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e + (2*I)*Sqrt[b 
]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(-Sqrt[b] + Sqrt[b + c*x]))] 
+ 30*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2 
*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sq 
rt[b + c*x])]))/(1920*c^(5/2)*e^6*x^(5/2)*(b + c*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {1162, 1231, 27, 1231, 27, 1269, 1091, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\int -\frac {\left (b d \left (16 c^2 d^2-22 b c e d+3 b^2 e^2\right )+(2 c d-b e) \left (16 c^2 d^2-16 b c e d-3 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)}dx}{8 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\int \frac {\left (b d \left (16 c^2 d^2-22 b c e d+3 b^2 e^2\right )+(2 c d-b e) \left (16 c^2 d^2-16 b c e d-3 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {-\frac {\int -\frac {b d \left (128 c^4 d^4-288 b c^3 e d^3+176 b^2 c^2 e^2 d^2-10 b^3 c e^3 d-3 b^4 e^4\right )+(2 c d-b e) \left (128 c^4 d^4-256 b c^3 e d^3+112 b^2 c^2 e^2 d^2+16 b^3 c e^3 d+3 b^4 e^4\right ) x}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\frac {\int \frac {b d \left (128 c^4 d^4-288 b c^3 e d^3+176 b^2 c^2 e^2 d^2-10 b^3 c e^3 d-3 b^4 e^4\right )+(2 c d-b e) \left (128 c^4 d^4-256 b c^3 e d^3+112 b^2 c^2 e^2 d^2+16 b^3 c e^3 d+3 b^4 e^4\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\frac {\frac {(2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {256 c^2 d^3 (c d-b e)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\frac {\frac {2 (2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {256 c^2 d^3 (c d-b e)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {256 c^2 d^3 (c d-b e)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\frac {\frac {512 c^2 d^3 (c d-b e)^3 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (b x+c x^2\right )^{5/2}}{5 e}-\frac {\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {256 c^2 d^{5/2} (c d-b e)^{5/2} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

Input:

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

Output:

(b*x + c*x^2)^(5/2)/(5*e) - (-1/24*((16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2 - 
 6*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(c*e^2) + (-1/4*((128*c^4*d^4 
 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e^4 - 2* 
c*e*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e - 3*b^2*e^2)*x)*Sqrt[b*x + c*x^ 
2])/(c*e^2) + ((2*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 112*b^2*c 
^2*d^2*e^2 + 16*b^3*c*d*e^3 + 3*b^4*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c* 
x^2]])/(Sqrt[c]*e) - (256*c^2*d^(5/2)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2* 
c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e)/(8*c*e^2) 
)/(16*c*e^2))/(2*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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1154

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

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 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 [A] (verified)

Time = 0.74 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {\left (-384 c^{4} e^{4} x^{4}-1008 b \,c^{3} e^{4} x^{3}+480 c^{4} d \,e^{3} x^{3}-744 b^{2} c^{2} e^{4} x^{2}+1360 b \,c^{3} d \,e^{3} x^{2}-640 c^{4} d^{2} e^{2} x^{2}-30 x \,b^{3} c \,e^{4}+1180 b^{2} c^{2} d \,e^{3} x -2080 b \,c^{3} d^{2} e^{2} x +960 c^{4} d^{3} e x +45 b^{4} e^{4}+150 d \,e^{3} b^{3} c -2640 d^{2} e^{2} b^{2} c^{2}+4320 d^{3} e b \,c^{3}-1920 d^{4} c^{4}\right ) x \left (c x +b \right )}{1920 c^{2} e^{5} \sqrt {x \left (c x +b \right )}}+\frac {\frac {\left (3 b^{5} e^{5}+10 b^{4} c d \,e^{4}+80 b^{3} d^{2} e^{3} c^{2}-480 b^{2} c^{3} d^{3} e^{2}+640 b \,c^{4} d^{4} e -256 d^{5} c^{5}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {256 d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) c^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{256 e^{5} c^{2}}\)	\(476\)
default	\(\frac {\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {5}{2}}}{5}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 e}-\frac {d \left (b e -c d \right ) \left (\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}\right )}{e^{2}}}{e}\)	\(926\)

Input:

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

Output:

-1/1920/c^2*(-384*c^4*e^4*x^4-1008*b*c^3*e^4*x^3+480*c^4*d*e^3*x^3-744*b^2 
*c^2*e^4*x^2+1360*b*c^3*d*e^3*x^2-640*c^4*d^2*e^2*x^2-30*b^3*c*e^4*x+1180* 
b^2*c^2*d*e^3*x-2080*b*c^3*d^2*e^2*x+960*c^4*d^3*e*x+45*b^4*e^4+150*b^3*c* 
d*e^3-2640*b^2*c^2*d^2*e^2+4320*b*c^3*d^3*e-1920*c^4*d^4)*x*(c*x+b)/e^5/(x 
*(c*x+b))^(1/2)+1/256/e^5/c^2*((3*b^5*e^5+10*b^4*c*d*e^4+80*b^3*c^2*d^2*e^ 
3-480*b^2*c^3*d^3*e^2+640*b*c^4*d^4*e-256*c^5*d^5)/e*ln((1/2*b+c*x)/c^(1/2 
)+(c*x^2+b*x)^(1/2))/c^(1/2)+256*d^3*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e- 
c^3*d^3)*c^2/e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c* 
d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e) 
-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e)))
 

Fricas [A] (verification not implemented)

Time = 2.44 (sec) , antiderivative size = 1544, normalized size of antiderivative = 4.06 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/3840*(15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3 
*c^2*d^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*sqrt(c)*log(2*c*x + b + 2*sqrt( 
c*x^2 + b*x)*sqrt(c)) - 3840*(c^5*d^4 - 2*b*c^4*d^3*e + b^2*c^3*d^2*e^2)*s 
qrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt 
(c*x^2 + b*x))/(e*x + d)) - 2*(384*c^5*e^5*x^4 + 1920*c^5*d^4*e - 4320*b*c 
^4*d^3*e^2 + 2640*b^2*c^3*d^2*e^3 - 150*b^3*c^2*d*e^4 - 45*b^4*c*e^5 - 48* 
(10*c^5*d*e^4 - 21*b*c^4*e^5)*x^3 + 8*(80*c^5*d^2*e^3 - 170*b*c^4*d*e^4 + 
93*b^2*c^3*e^5)*x^2 - 10*(96*c^5*d^3*e^2 - 208*b*c^4*d^2*e^3 + 118*b^2*c^3 
*d*e^4 - 3*b^3*c^2*e^5)*x)*sqrt(c*x^2 + b*x))/(c^3*e^6), -1/3840*(7680*(c^ 
5*d^4 - 2*b*c^4*d^3*e + b^2*c^3*d^2*e^2)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt( 
-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) + 15*(256*c^5*d^5 - 640*b 
*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 - 3 
*b^5*e^5)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(384*c^ 
5*e^5*x^4 + 1920*c^5*d^4*e - 4320*b*c^4*d^3*e^2 + 2640*b^2*c^3*d^2*e^3 - 1 
50*b^3*c^2*d*e^4 - 45*b^4*c*e^5 - 48*(10*c^5*d*e^4 - 21*b*c^4*e^5)*x^3 + 8 
*(80*c^5*d^2*e^3 - 170*b*c^4*d*e^4 + 93*b^2*c^3*e^5)*x^2 - 10*(96*c^5*d^3* 
e^2 - 208*b*c^4*d^2*e^3 + 118*b^2*c^3*d*e^4 - 3*b^3*c^2*e^5)*x)*sqrt(c*x^2 
 + b*x))/(c^3*e^6), 1/1920*(15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^ 
3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*sqrt(-c)*arct 
an(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) + 1920*(c^5*d^4 - 2*b*c^4*d^3*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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