\(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^{3/2}} \, dx\) [414]

Optimal result

Integrand size = 23, antiderivative size = 395 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}+\frac {2 (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {4 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2*(e*x+d)^(5/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^(1/2)+2/3*(-b*e+2*c*d)*(8*b^2*e^2-3*b*c*d*e+3*c^2*d^2)*E
llipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(3/2)/c^(5/2)/
(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-4/3*d*(-b*e+c*d)*(2*b^2*e^2-3*b*c*d*e+3*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(
-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(3/2)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+b*x
)^(1/2)+2*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c*x^2+b*x)^(1/2)/b^2/c+4/3*e*(2*b^2*e^2-3*b*c*d*e+3*c^2*d^2)*(e*x+d)^(
1/2)*(c*x^2+b*x)^(1/2)/b^2/c^2

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 395, 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 = {752, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {4 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {4 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 c^2}-\frac {2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{b^2 c} \]

[In]

Int[(d + e*x)^(7/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (4*e*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2
)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (2*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(b^2*c)
+ (2*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcS
in[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (4*
d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2*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)])/(3*(-b)^(3/2)*c^(5/2)*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 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && LtQ[p, -1] && GtQ[m, 1] && 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 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}-\frac {2 \int \frac {(d+e x)^{3/2} \left (-\frac {5}{2} b d e-\frac {5}{2} e (2 c d-b e) x\right )}{\sqrt {b x+c x^2}} \, dx}{b^2} \\ & = -\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}-\frac {4 \int \frac {\sqrt {d+e x} \left (-\frac {5}{4} b d e (3 c d+b e)-\frac {5}{2} e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{5 b^2 c} \\ & = -\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}-\frac {8 \int \frac {-\frac {5}{8} b d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )-\frac {5}{8} e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 b^2 c^2} \\ & = -\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}-\frac {\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 c^2}+\frac {\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}-\frac {\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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}{3 b^2 c^2 \sqrt {b x+c x^2}}+\frac {\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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}{3 b^2 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}+\frac {\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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}{3 b^2 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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}{3 b^2 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {4 e \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{b^2 c}+\frac {2 (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 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 )}{3 (-b)^{3/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {4 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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 )}{3 (-b)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.89 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (b (d+e x) \left (3 (c d-b e)^3 x+3 c^2 d^3 (b+c x)-b^2 e^3 x (b+c x)\right )-\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (6 c^3 d^3-9 b c^2 d^2 e+19 b^2 c d e^2-8 b^3 e^3\right ) (b+c x) (d+e x)+i b e \left (6 c^3 d^3-9 b c^2 d^2 e+19 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (3 c^3 d^3-18 b c^2 d^2 e+23 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^3 c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[In]

Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(b*(d + e*x)*(3*(c*d - b*e)^3*x + 3*c^2*d^3*(b + c*x) - b^2*e^3*x*(b + c*x)) - Sqrt[b/c]*(Sqrt[b/c]*(6*c^3
*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19
*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]],
(c*d)/(b*e)] - I*b*e*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*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)])))/(3*b^3*c^2*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. \(699\) vs. \(2(339)=678\).

Time = 2.01 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.77

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d^{3}}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (c e \,x^{2}+c d x \right ) \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{b^{2} c^{3} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 e^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c^{2}}+\frac {2 \left (\frac {e^{2} \left (b^{2} e^{2}-4 b c d e +6 c^{2} d^{2}\right )}{c^{3}}-\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \left (b e -c d \right )}{c^{3} b^{2}}-\frac {d \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{c^{2} b^{2}}-\frac {e^{3} b d}{3 c^{2}}\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 (-\frac {e^{3} \left (b e -4 c d \right )}{c^{2}}+\frac {d^{3} c e}{b^{2}}-\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) e}{c^{2} b^{2}}-\frac {2 e^{3} \left (b e +c d \right )}{3 c^{2}}\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}}\)	\(700\)
default	\(\frac {2 \left (4 \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}-10 \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}+12 \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 -6 \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}+8 \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}-27 \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}+28 \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}-15 \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 +6 \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}+b^{2} c^{3} e^{4} x^{3}+4 b^{3} c^{2} e^{4} x^{2}-8 b^{2} c^{3} d \,e^{3} x^{2}+9 b \,c^{4} d^{2} e^{2} x^{2}-6 c^{5} d^{3} e \,x^{2}+4 b^{3} c^{2} d \,e^{3} x -9 b^{2} c^{3} d^{2} e^{2} x +6 b \,c^{4} d^{3} e x -6 x \,c^{5} d^{4}-3 b \,d^{4} c^{4}\right ) \sqrt {x \left (c x +b \right )}}{3 x \,c^{4} \left (c x +b \right ) b^{2} \sqrt {e x +d}}\)	\(863\)

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x)^(3/2),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*(c*e*x^2+b*e*x+c*d*x+b*d)*d^3/b^2/(x*(c*e*x^2+b*
e*x+c*d*x+b*d))^(1/2)+2*(c*e*x^2+c*d*x)*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/b^2/c^3/((1/c*b+x)*(c*e*
x^2+c*d*x))^(1/2)+2/3*e^3/c^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(e^2*(b^2*e^2-4*b*c*d*e+6*c^2*d^2)/c^3-(
b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/c^3*(b*e-c*d)/b^2-1/c^2*d*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^
3*d^3)/b^2-1/3*e^3/c^2*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*(-1/c^2*e^3*(b*e-4*c*
d)+d^3*c*e/b^2-(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/b^2-2/3*e^3/c^2*(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)*Elli
pticE(((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 = 0.14 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.54 \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e - 17 \, b^{2} c^{3} d^{2} e^{2} + 23 \, b^{3} c^{2} d e^{3} - 8 \, b^{4} c e^{4}\right )} x^{2} + {\left (6 \, b c^{4} d^{4} - 12 \, b^{2} c^{3} d^{3} e - 17 \, b^{3} c^{2} d^{2} e^{2} + 23 \, b^{4} c d e^{3} - 8 \, b^{5} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (6 \, c^{5} d^{3} e - 9 \, b c^{4} d^{2} e^{2} + 19 \, b^{2} c^{3} d e^{3} - 8 \, b^{3} c^{2} e^{4}\right )} x^{2} + {\left (6 \, b c^{4} d^{3} e - 9 \, b^{2} c^{3} d^{2} e^{2} + 19 \, b^{3} c^{2} d e^{3} - 8 \, b^{4} c e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (b^{2} c^{3} e^{4} x^{2} - 3 \, b c^{4} d^{3} e - {\left (6 \, c^{5} d^{3} e - 9 \, b c^{4} d^{2} e^{2} + 9 \, b^{2} c^{3} d e^{3} - 4 \, b^{3} c^{2} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{2} c^{5} e x^{2} + b^{3} c^{4} e x\right )}} \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

-2/9*(((6*c^5*d^4 - 12*b*c^4*d^3*e - 17*b^2*c^3*d^2*e^2 + 23*b^3*c^2*d*e^3 - 8*b^4*c*e^4)*x^2 + (6*b*c^4*d^4 -
12*b^2*c^3*d^3*e - 17*b^3*c^2*d^2*e^2 + 23*b^4*c*d*e^3 - 8*b^5*e^4)*x)*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)) + 3*((6*c^5*d^3*e - 9*b*c^4*d^2*e^2 + 19*b^2*c^3*d*e^3 - 8*b^3*c^2*e^4)*x^2 +
(6*b*c^4*d^3*e - 9*b^2*c^3*d^2*e^2 + 19*b^3*c^2*d*e^3 - 8*b^4*c*e^4)*x)*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), wei
erstrassPInverse(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*(b^2*c^3*e^4*x^2 - 3*b*c^4*d^3*e - (6*c^5*d^3*e
- 9*b*c^4*d^2*e^2 + 9*b^2*c^3*d*e^3 - 4*b^3*c^2*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(b^2*c^5*e*x^2 + b^3
*c^4*e*x)

Sympy [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((d + e*x)**(7/2)/(x*(b + c*x))**(3/2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]

[In]

int((d + e*x)^(7/2)/(b*x + c*x^2)^(3/2),x)

[Out]

int((d + e*x)^(7/2)/(b*x + c*x^2)^(3/2), x)