\(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\) [405]

Optimal result

Integrand size = 23, antiderivative size = 570 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d^2 e^6 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/63*(d*(-2*b^2*e^2-11*b*c*d*e+16*c^2*d^2)+e*(3*b^2*e^2-26*b*c*d*e+26*c^2*d^2)*x)*(c*x^2+b*x)^(3/2)/d/e^3/(-b
*e+c*d)/(e*x+d)^(7/2)-2/9*(c*x^2+b*x)^(5/2)/e/(e*x+d)^(9/2)+4/63*(-b^4*e^4-7*b^3*c*d*e^3+135*b^2*c^2*d^2*e^2-2
56*b*c^3*d^3*e+128*c^4*d^4)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(
1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d^2/e^6/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/63*(-b*e+2*c*d)*(-b^2*e^
2-128*b*c*d*e+128*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(1
+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/d/e^6/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/63*(c*d^2*(-b^3*e^3+111*b^2*c
*d*e^2-240*b*c^2*d^2*e+128*c^3*d^3)+e*(-2*b^4*e^4-11*b^3*c*d*e^3+171*b^2*c^2*d^2*e^2-320*b*c^3*d^3*e+160*c^4*d
^4)*x)*(c*x^2+b*x)^(1/2)/d^2/e^5/(-b*e+c*d)^2/(e*x+d)^(3/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 570, 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 = {746, 824, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{63 d e^6 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d^2 e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(c*d^2*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3) + e*(160*c^4*d^4 - 320*b*c^3*d^3*e + 17
1*b^2*c^2*d^2*e^2 - 11*b^3*c*d*e^3 - 2*b^4*e^4)*x)*Sqrt[b*x + c*x^2])/(63*d^2*e^5*(c*d - b*e)^2*(d + e*x)^(3/2
)) - (2*(d*(16*c^2*d^2 - 11*b*c*d*e - 2*b^2*e^2) + e*(26*c^2*d^2 - 26*b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3
/2))/(63*d*e^3*(c*d - b*e)*(d + e*x)^(7/2)) - (2*(b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (4*Sqrt[-b]*Sqrt
[c]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d^2*e^6*(c*d - b*e)^2*Sqrt[1 + (
e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e - b^2*e^2)*Sqrt[x]*S
qrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d*e^6*(c*d
- b*e)*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 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
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 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e} \\ & = -\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {2 \int \frac {\left (-\frac {1}{2} b \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )-\frac {1}{2} c \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx}{21 d e^3 (c d-b e)} \\ & = -\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \int \frac {\frac {1}{4} b c d \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+\frac {1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 d^2 e^5 (c d-b e)^2} \\ & = -\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 d e^6 (c d-b e)}+\frac {\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{63 d^2 e^6 (c d-b e)^2} \\ & = -\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-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}{63 d e^6 (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{63 d^2 e^6 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\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}{63 d^2 e^6 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-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}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\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 )}{63 d^2 e^6 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e-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 )}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.80 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.07 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (7 d^4 (c d-b e)^4-19 d^3 (c d-b e)^2 \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) (d+e x)+d^2 (c d-b e)^2 \left (88 c^2 d^2-88 b c d e+15 b^2 e^2\right ) (d+e x)^2-d (c d-b e) \left (122 c^3 d^3-183 b c^2 d^2 e+63 b^2 c d e^2-b^3 e^3\right ) (d+e x)^3+\left (193 c^4 d^4-386 b c^3 d^3 e+207 b^2 c^2 d^2 e^2-14 b^3 c d e^3-2 b^4 e^4\right ) (d+e x)^4\right )-\sqrt {\frac {b}{c}} c (d+e x)^4 \left (-2 \sqrt {\frac {b}{c}} \left (-128 c^4 d^4+256 b c^3 d^3 e-135 b^2 c^2 d^2 e^2+7 b^3 c d e^3+b^4 e^4\right ) (b+c x) (d+e x)+2 i b e \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (128 c^4 d^4-272 b c^3 d^3 e+159 b^2 c^2 d^2 e^2-13 b^3 c d e^3-2 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{63 b d^2 e^6 (c d-b e)^2 x^3 (b+c x)^3 (d+e x)^{9/2}} \]

[In]

Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(7*d^4*(c*d - b*e)^4 - 19*d^3*(c*d - b*e)^2*(2*c^2*d^2 - 3*b*c*d*e +
b^2*e^2)*(d + e*x) + d^2*(c*d - b*e)^2*(88*c^2*d^2 - 88*b*c*d*e + 15*b^2*e^2)*(d + e*x)^2 - d*(c*d - b*e)*(122
*c^3*d^3 - 183*b*c^2*d^2*e + 63*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^3 + (193*c^4*d^4 - 386*b*c^3*d^3*e + 207*b^2*
c^2*d^2*e^2 - 14*b^3*c*d*e^3 - 2*b^4*e^4)*(d + e*x)^4) - Sqrt[b/c]*c*(d + e*x)^4*(-2*Sqrt[b/c]*(-128*c^4*d^4 +
256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 + b^4*e^4)*(b + c*x)*(d + e*x) + (2*I)*b*e*(128*c^4*d^4
- 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*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*(128*c^4*d^4 - 272*b*c^3*d^3*e + 159*b^2*c^2*d^
2*e^2 - 13*b^3*c*d*e^3 - 2*b^4*e^4)*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)])))/(63*b*d^2*e^6*(c*d - b*e)^2*x^3*(b + c*x)^3*(d + e*x)^(9/2))

Maple [A] (verified)

Time = 4.20 (sec) , antiderivative size = 995, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{9 e^{10} \left (x +\frac {d}{e}\right )^{5}}+\frac {38 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{63 e^{9} \left (x +\frac {d}{e}\right )^{4}}-\frac {2 \left (15 b^{2} e^{2}-88 b c d e +88 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{63 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (b^{3} e^{3}-63 b^{2} d \,e^{2} c +183 b \,c^{2} d^{2} e -122 c^{3} d^{3}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{63 d \left (b e -c d \right ) e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}\right )}{63 d^{2} \left (b e -c d \right )^{2} e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c^{2} \left (3 b e -5 c d \right )}{e^{6}}+\frac {c \left (b^{3} e^{3}-63 b^{2} d \,e^{2} c +183 b \,c^{2} d^{2} e -122 c^{3} d^{3}\right )}{63 d \left (b e -c d \right ) e^{6}}+\frac {2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}}{63 e^{6} \left (b e -c d \right ) d^{2}}-\frac {b \left (2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}\right )}{63 e^{5} d^{2} \left (b e -c d \right )^{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 {c^{3}}{e^{5}}-\frac {c \left (2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}\right )}{63 e^{5} d^{2} \left (b e -c d \right )^{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 {e x +d}\, x \left (c x +b \right )}\)	\(995\)
default	\(\text {Expression too large to display}\)	\(5005\)

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)

[Out]

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(x*(e*x+d)*(c*x+b))^(1/2)/x/(c*x+b)*(-2/9*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^
10*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^5+38/63*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^9*(c*e*x^3+b*e*x^2+
c*d*x^2+b*d*x)^(1/2)/(x+d/e)^4-2/63*(15*b^2*e^2-88*b*c*d*e+88*c^2*d^2)/e^8*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/
2)/(x+d/e)^3+2/63*(b^3*e^3-63*b^2*c*d*e^2+183*b*c^2*d^2*e-122*c^3*d^3)/d/(b*e-c*d)/e^7*(c*e*x^3+b*e*x^2+c*d*x^
2+b*d*x)^(1/2)/(x+d/e)^2+2/63*(c*e*x^2+b*e*x)/d^2/(b*e-c*d)^2/e^6*(2*b^4*e^4+14*b^3*c*d*e^3-207*b^2*c^2*d^2*e^
2+386*b*c^3*d^3*e-193*c^4*d^4)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(c^2*(3*b*e-5*c*d)/e^6+1/63*c*(b^3*e^3-63*b^2
*c*d*e^2+183*b*c^2*d^2*e-122*c^3*d^3)/d/(b*e-c*d)/e^6+1/63/e^6/(b*e-c*d)*(2*b^4*e^4+14*b^3*c*d*e^3-207*b^2*c^2
*d^2*e^2+386*b*c^3*d^3*e-193*c^4*d^4)/d^2-1/63*b/e^5/d^2/(b*e-c*d)^2*(2*b^4*e^4+14*b^3*c*d*e^3-207*b^2*c^2*d^2
*e^2+386*b*c^3*d^3*e-193*c^4*d^4))/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*(c^3/e^5-1/63/
e^5*c*(2*b^4*e^4+14*b^3*c*d*e^3-207*b^2*c^2*d^2*e^2+386*b*c^3*d^3*e-193*c^4*d^4)/d^2/(b*e-c*d)^2)/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))))

Fricas [C] (verification not implemented)

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

Time = 0.26 (sec) , antiderivative size = 1675, normalized size of antiderivative = 2.94 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

-2/189*((256*c^5*d^10 - 640*b*c^4*d^9*e + 478*b^2*c^3*d^8*e^2 - 77*b^3*c^2*d^7*e^3 - 13*b^4*c*d^6*e^4 - 2*b^5*
d^5*e^5 + (256*c^5*d^5*e^5 - 640*b*c^4*d^4*e^6 + 478*b^2*c^3*d^3*e^7 - 77*b^3*c^2*d^2*e^8 - 13*b^4*c*d*e^9 - 2
*b^5*e^10)*x^5 + 5*(256*c^5*d^6*e^4 - 640*b*c^4*d^5*e^5 + 478*b^2*c^3*d^4*e^6 - 77*b^3*c^2*d^3*e^7 - 13*b^4*c*
d^2*e^8 - 2*b^5*d*e^9)*x^4 + 10*(256*c^5*d^7*e^3 - 640*b*c^4*d^6*e^4 + 478*b^2*c^3*d^5*e^5 - 77*b^3*c^2*d^4*e^
6 - 13*b^4*c*d^3*e^7 - 2*b^5*d^2*e^8)*x^3 + 10*(256*c^5*d^8*e^2 - 640*b*c^4*d^7*e^3 + 478*b^2*c^3*d^6*e^4 - 77
*b^3*c^2*d^5*e^5 - 13*b^4*c*d^4*e^6 - 2*b^5*d^3*e^7)*x^2 + 5*(256*c^5*d^9*e - 640*b*c^4*d^8*e^2 + 478*b^2*c^3*
d^7*e^3 - 77*b^3*c^2*d^6*e^4 - 13*b^4*c*d^5*e^5 - 2*b^5*d^4*e^6)*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)) + 6*(128*c^5*d^9*e - 256*b*c^4*d^8*e^2 + 135*b^2*c^3*d^7*e^3 - 7*b^3*c^2*d^6*e^4
- b^4*c*d^5*e^5 + (128*c^5*d^4*e^6 - 256*b*c^4*d^3*e^7 + 135*b^2*c^3*d^2*e^8 - 7*b^3*c^2*d*e^9 - b^4*c*e^10)*x
^5 + 5*(128*c^5*d^5*e^5 - 256*b*c^4*d^4*e^6 + 135*b^2*c^3*d^3*e^7 - 7*b^3*c^2*d^2*e^8 - b^4*c*d*e^9)*x^4 + 10*
(128*c^5*d^6*e^4 - 256*b*c^4*d^5*e^5 + 135*b^2*c^3*d^4*e^6 - 7*b^3*c^2*d^3*e^7 - b^4*c*d^2*e^8)*x^3 + 10*(128*
c^5*d^7*e^3 - 256*b*c^4*d^6*e^4 + 135*b^2*c^3*d^5*e^5 - 7*b^3*c^2*d^4*e^6 - b^4*c*d^3*e^7)*x^2 + 5*(128*c^5*d^
8*e^2 - 256*b*c^4*d^7*e^3 + 135*b^2*c^3*d^6*e^4 - 7*b^3*c^2*d^5*e^5 - b^4*c*d^4*e^6)*x)*sqrt(c*e)*weierstrassZ
eta(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*(128*c^5*d^8*e^2 - 240*b*c^4*d^7
*e^3 + 111*b^2*c^3*d^6*e^4 - b^3*c^2*d^5*e^5 + (193*c^5*d^4*e^6 - 386*b*c^4*d^3*e^7 + 207*b^2*c^3*d^2*e^8 - 14
*b^3*c^2*d*e^9 - 2*b^4*c*e^10)*x^4 + (650*c^5*d^5*e^5 - 1239*b*c^4*d^4*e^6 + 582*b^2*c^3*d^3*e^7 + 8*b^3*c^2*d
^2*e^8 - 9*b^4*c*d*e^9)*x^3 + (880*c^5*d^6*e^4 - 1665*b*c^4*d^5*e^5 + 783*b^2*c^3*d^4*e^6 - 10*b^3*c^2*d^3*e^7
)*x^2 + (544*c^5*d^7*e^3 - 1024*b*c^4*d^6*e^4 + 477*b^2*c^3*d^5*e^5 - 5*b^3*c^2*d^4*e^6)*x)*sqrt(c*x^2 + b*x)*
sqrt(e*x + d))/(c^3*d^9*e^7 - 2*b*c^2*d^8*e^8 + b^2*c*d^7*e^9 + (c^3*d^4*e^12 - 2*b*c^2*d^3*e^13 + b^2*c*d^2*e
^14)*x^5 + 5*(c^3*d^5*e^11 - 2*b*c^2*d^4*e^12 + b^2*c*d^3*e^13)*x^4 + 10*(c^3*d^6*e^10 - 2*b*c^2*d^5*e^11 + b^
2*c*d^4*e^12)*x^3 + 10*(c^3*d^7*e^9 - 2*b*c^2*d^6*e^10 + b^2*c*d^5*e^11)*x^2 + 5*(c^3*d^8*e^8 - 2*b*c^2*d^7*e^
9 + b^2*c*d^6*e^10)*x)

Sympy [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]

[In]

integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(11/2), x)

Maxima [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2), x)

Giac [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]

[In]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x)

[Out]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(11/2), x)