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

Optimal result

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

[Out]

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)*(2
4*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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {756, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c} \]

[In]

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

[Out]

(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])/(105*c^3) + (12*e*(2*c*d - b*e)*(
d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*e*(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*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)])/(105*c^(7/2)*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[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(7/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 113

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[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] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[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] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 729

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (7 c d-b e)+3 e (2 c d-b e) x\right )}{\sqrt {b x+c x^2}} \, dx}{7 c} \\ & = \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 {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} d \left (35 c^2 d^2-17 b c d e+6 b^2 e^2\right )+\frac {1}{4} e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{35 c^2} \\ & = \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 {8 \int \frac {\frac {1}{8} d (7 c d-3 b e) \left (15 c^2 d^2-11 b c d e+8 b^2 e^2\right )+e (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^3} \\ & = \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 {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{105 c^3}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^3} \\ & = \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 {\left (8 (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 {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{105 c^3 \sqrt {b x+c x^2}}-\frac {\left (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 {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{105 c^3 \sqrt {b x+c x^2}} \\ & = \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 {\left (8 (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}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{105 c^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (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}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{105 c^3 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

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

[In]

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

[Out]

(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)*Sqrt[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*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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 )}\)	\(918\)

[In]

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

[Out]

(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)*Elliptic
E(((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))))

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

[In]

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

[Out]

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)*weierstr
assPInverse(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)*s
qrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^5*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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