3.5.6 \(\int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx\) [406]

3.5.6.1 Optimal result

Integrand size = 23, antiderivative size = 379 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {16 \sqrt {-b} (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

output

16/105*(-b*e+2*c*d)*(6*b^2*e^2-11*b*c*d*e+11*c^2*d^2)*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)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/105*d*(-b*e+c*d)*(24* 
b^2*e^2-71*b*c*d*e+71*c^2*d^2)*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*x 
+d)^(1/2)/(c*x^2+b*x)^(1/2)+12/35*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c*x^2+b*x) 
^(1/2)/c^2+2/7*e*(e*x+d)^(5/2)*(c*x^2+b*x)^(1/2)/c+2/105*e*(24*b^2*e^2-71* 
b*c*d*e+71*c^2*d^2)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^3
 

3.5.6.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.22 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x} \left (\frac {8 \left (22 c^3 d^3-33 b c^2 d^2 e+23 b^2 c d e^2-6 b^3 e^3\right ) (b+c x) (d+e x)}{c \sqrt {x}}+e \sqrt {x} (b+c x) (d+e x) \left (24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+8 i \sqrt {\frac {b}{c}} e \left (22 c^3 d^3-33 b c^2 d^2 e+23 b^2 c d e^2-6 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\frac {i \sqrt {\frac {b}{c}} \left (105 c^4 d^4-298 b c^3 d^3 e+353 b^2 c^2 d^2 e^2-208 b^3 c d e^3+48 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b}\right )}{105 c^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

input

Integrate[(d + e*x)^(7/2)/Sqrt[b*x + c*x^2],x]
 

output

(2*Sqrt[x]*((8*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)* 
(b + c*x)*(d + e*x))/(c*Sqrt[x]) + e*Sqrt[x]*(b + c*x)*(d + e*x)*(24*b^2*e 
^2 - b*c*e*(89*d + 18*e*x) + c^2*(122*d^2 + 66*d*e*x + 15*e^2*x^2)) + (8*I 
)*Sqrt[b/c]*e*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*S 
qrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x] 
], (c*d)/(b*e)] + (I*Sqrt[b/c]*(105*c^4*d^4 - 298*b*c^3*d^3*e + 353*b^2*c^ 
2*d^2*e^2 - 208*b^3*c*d*e^3 + 48*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e* 
x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(105*c^3*S 
qrt[x*(b + c*x)]*Sqrt[d + e*x])
 

3.5.6.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {1166, 27, 1236, 27, 1236, 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.

\(\displaystyle \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {(d+e x)^{3/2} (d (7 c d-b e)+6 e (2 c d-b e) x)}{2 \sqrt {c x^2+b x}}dx}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(d+e x)^{3/2} (d (7 c d-b e)+6 e (2 c d-b e) x)}{\sqrt {c x^2+b x}}dx}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {d+e x} \left (d \left (35 c^2 d^2-17 b c e d+6 b^2 e^2\right )+e \left (71 c^2 d^2-71 b c e d+24 b^2 e^2\right ) x\right )}{2 \sqrt {c x^2+b x}}dx}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {d+e x} \left (d \left (35 c^2 d^2-17 b c e d+6 b^2 e^2\right )+e \left (71 c^2 d^2-71 b c e d+24 b^2 e^2\right ) x\right )}{\sqrt {c x^2+b x}}dx}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {\frac {2 \int \frac {d (7 c d-3 b e) \left (15 c^2 d^2-11 b c e d+8 b^2 e^2\right )+8 e (2 c d-b e) \left (11 c^2 d^2-11 b c e d+6 b^2 e^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {d (7 c d-3 b e) \left (15 c^2 d^2-11 b c e d+8 b^2 e^2\right )+8 e (2 c d-b e) \left (11 c^2 d^2-11 b c e d+6 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {8 (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx-d (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {\frac {\frac {\frac {8 \sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{\sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {\frac {\frac {\frac {8 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{\sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{\sqrt {b x+c x^2} \sqrt {d+e x}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c}\)

input

Int[(d + e*x)^(7/2)/Sqrt[b*x + c*x^2],x]
 

output

(2*e*(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*c) + ((12*e*(2*c*d - b*e)*(d + 
e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*c) + ((2*e*(71*c^2*d^2 - 71*b*c*d*e + 24* 
b^2*e^2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + ((16*Sqrt[-b]*(2*c*d - b 
*e)*(11*c^2*d^2 - 11*b*c*d*e + 6*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d 
 + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[ 
c]*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(71*c^ 
2*d^2 - 71*b*c*d*e + 24*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/ 
d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*Sq 
rt[d + e*x]*Sqrt[b*x + c*x^2]))/(3*c))/(5*c))/(7*c)
 

3.5.6.3.1 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 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[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 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]
 

3.5.6.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(696\) vs. \(2(319)=638\).

Time = 2.05 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.84

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 c}+\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 c e}+\frac {2 \left (6 d^{2} e^{2}-\frac {5 e^{3} b d}{7 c}-\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (d^{4}-\frac {\left (6 d^{2} e^{2}-\frac {5 e^{3} b d}{7 c}-\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) b d}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (4 d^{3} e -\frac {3 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) b d}{5 c e}-\frac {2 \left (6 d^{2} e^{2}-\frac {5 e^{3} b d}{7 c}-\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \left (b e +c d \right )}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(697\)
default	\(\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (15 c^{5} e^{4} x^{5}+24 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} c d \,e^{3}-95 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c^{2} d^{2} e^{2}+142 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{3} d^{3} e -71 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{4} d^{4}+48 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{5} e^{4}-232 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} c d \,e^{3}+448 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c^{2} d^{2} e^{2}-440 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{3} d^{3} e +176 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{4} d^{4}-3 b \,c^{4} e^{4} x^{4}+81 c^{5} d \,e^{3} x^{4}+6 b^{2} c^{3} e^{4} x^{3}-26 b \,c^{4} d \,e^{3} x^{3}+188 c^{5} d^{2} e^{2} x^{3}+24 b^{3} c^{2} e^{4} x^{2}-83 b^{2} c^{3} d \,e^{3} x^{2}+99 b \,c^{4} d^{2} e^{2} x^{2}+122 c^{5} d^{3} e \,x^{2}+24 b^{3} c^{2} d \,e^{3} x -89 b^{2} c^{3} d^{2} e^{2} x +122 b \,c^{4} d^{3} e x \right )}{105 c^{5} x \left (c e \,x^{2}+b e x +c d x +b d \right )}\)	\(918\)

input

int((e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
 

output

(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/7*e^3/c*x^2*( 
c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d)) 
/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(6*d^2*e^2-5/7*e^3/c*b*d- 
2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e* 
x^2+c*d*x^2+b*d*x)^(1/2)+2*(d^4-1/3*(6*d^2*e^2-5/7*e^3/c*b*d-2/5*(4*d*e^3- 
2/7*e^3/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*b*d)/c*b*((1/c*b+x)*c/b)^( 
1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+ 
b*d*x)^(1/2)*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))+ 
2*(4*d^3*e-3/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*b*d-2/3*(6*d^2*e^2-5/ 
7*e^3/c*b*d-2/5*(4*d*e^3-2/7*e^3/c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*( 
b*e+c*d))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^ 
(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-1/c*b+d/e)*EllipticE(((1/c* 
b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF(((1/c*b+x)*c/b) 
^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))))
 

3.5.6.5 Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (139 \, c^{4} d^{4} - 278 \, b c^{3} d^{3} e + 347 \, b^{2} c^{2} d^{2} e^{2} - 208 \, b^{3} c d e^{3} + 48 \, b^{4} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 24 \, {\left (22 \, c^{4} d^{3} e - 33 \, b c^{3} d^{2} e^{2} + 23 \, b^{2} c^{2} d e^{3} - 6 \, b^{3} c e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 122 \, c^{4} d^{2} e^{2} - 89 \, b c^{3} d e^{3} + 24 \, b^{2} c^{2} e^{4} + 6 \, {\left (11 \, c^{4} d e^{3} - 3 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{315 \, c^{5} e} \]

input

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
 

output

2/315*((139*c^4*d^4 - 278*b*c^3*d^3*e + 347*b^2*c^2*d^2*e^2 - 208*b^3*c*d* 
e^3 + 48*b^4*e^4)*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)) - 24*(22*c^4*d^3*e - 33 
*b*c^3*d^2*e^2 + 23*b^2*c^2*d*e^3 - 6*b^3*c*e^4)*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*( 
15*c^4*e^4*x^2 + 122*c^4*d^2*e^2 - 89*b*c^3*d*e^3 + 24*b^2*c^2*e^4 + 6*(11 
*c^4*d*e^3 - 3*b*c^3*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^5*e)
 

3.5.6.6 Sympy [F]

\[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]

input

integrate((e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)
 

output

Integral((d + e*x)**(7/2)/sqrt(x*(b + c*x)), x)
 

3.5.6.7 Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]

input

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")
 

output

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x), x)
 

3.5.6.8 Giac [F]

\[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]

input

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")
 

output

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x), x)
 

3.5.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {c\,x^2+b\,x}} \,d x \]

input

int((d + e*x)^(7/2)/(b*x + c*x^2)^(1/2),x)
 

output

int((d + e*x)^(7/2)/(b*x + c*x^2)^(1/2), x)