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

Optimal. Leaf size=570 \[ -\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d^2 e^6 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[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)*(
c*x/b+1)^(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)*(c
*x/b+1)^(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]
time = 0.45, antiderivative size = 570, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {746, 824, 857, 729, 113, 111, 118, 117} \begin {gather*} -\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 ) F\left (\text {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 (\text {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}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains complex when optimal does not.
time = 22.31, size = 610, normalized size = 1.07 \begin {gather*} -\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 \sinh ^{-1}\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} F\left (i \sinh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5004\) vs. \(2(510)=1020\).
time = 0.47, size = 5005, normalized size = 8.78

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 +d^{2} c^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{9 e^{10} \left (x +\frac {d}{e}\right )^{5}}+\frac {38 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{63 e^{9} \left (x +\frac {d}{e}\right )^{4}}-\frac {2 \left (15 b^{2} e^{2}-88 b c d e +88 d^{2} c^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{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}+x b d}}{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}}\, \EllipticF \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}+x b d}}+\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 ) \EllipticE \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 \EllipticF \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}+x b d}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(995\)
default	\(\text {Expression too large to display}\)	\(5005\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/(x*e + d)^(11/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.84, size = 1642, normalized size = 2.88 \begin {gather*} -\frac {2 \, {\left ({\left (256 \, c^{5} d^{10} - 2 \, b^{5} x^{5} e^{10} - {\left (13 \, b^{4} c d x^{5} + 10 \, b^{5} d x^{4}\right )} e^{9} - {\left (77 \, b^{3} c^{2} d^{2} x^{5} + 65 \, b^{4} c d^{2} x^{4} + 20 \, b^{5} d^{2} x^{3}\right )} e^{8} + {\left (478 \, b^{2} c^{3} d^{3} x^{5} - 385 \, b^{3} c^{2} d^{3} x^{4} - 130 \, b^{4} c d^{3} x^{3} - 20 \, b^{5} d^{3} x^{2}\right )} e^{7} - 10 \, {\left (64 \, b c^{4} d^{4} x^{5} - 239 \, b^{2} c^{3} d^{4} x^{4} + 77 \, b^{3} c^{2} d^{4} x^{3} + 13 \, b^{4} c d^{4} x^{2} + b^{5} d^{4} x\right )} e^{6} + {\left (256 \, c^{5} d^{5} x^{5} - 3200 \, b c^{4} d^{5} x^{4} + 4780 \, b^{2} c^{3} d^{5} x^{3} - 770 \, b^{3} c^{2} d^{5} x^{2} - 65 \, b^{4} c d^{5} x - 2 \, b^{5} d^{5}\right )} e^{5} + {\left (1280 \, c^{5} d^{6} x^{4} - 6400 \, b c^{4} d^{6} x^{3} + 4780 \, b^{2} c^{3} d^{6} x^{2} - 385 \, b^{3} c^{2} d^{6} x - 13 \, b^{4} c d^{6}\right )} e^{4} + {\left (2560 \, c^{5} d^{7} x^{3} - 6400 \, b c^{4} d^{7} x^{2} + 2390 \, b^{2} c^{3} d^{7} x - 77 \, b^{3} c^{2} d^{7}\right )} e^{3} + 2 \, {\left (1280 \, c^{5} d^{8} x^{2} - 1600 \, b c^{4} d^{8} x + 239 \, b^{2} c^{3} d^{8}\right )} e^{2} + 640 \, {\left (2 \, c^{5} d^{9} x - b c^{4} d^{9}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{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 )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 6 \, {\left (128 \, c^{5} d^{9} e - b^{4} c x^{5} e^{10} - {\left (7 \, b^{3} c^{2} d x^{5} + 5 \, b^{4} c d x^{4}\right )} e^{9} + 5 \, {\left (27 \, b^{2} c^{3} d^{2} x^{5} - 7 \, b^{3} c^{2} d^{2} x^{4} - 2 \, b^{4} c d^{2} x^{3}\right )} e^{8} - {\left (256 \, b c^{4} d^{3} x^{5} - 675 \, b^{2} c^{3} d^{3} x^{4} + 70 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{4} c d^{3} x^{2}\right )} e^{7} + {\left (128 \, c^{5} d^{4} x^{5} - 1280 \, b c^{4} d^{4} x^{4} + 1350 \, b^{2} c^{3} d^{4} x^{3} - 70 \, b^{3} c^{2} d^{4} x^{2} - 5 \, b^{4} c d^{4} x\right )} e^{6} + {\left (640 \, c^{5} d^{5} x^{4} - 2560 \, b c^{4} d^{5} x^{3} + 1350 \, b^{2} c^{3} d^{5} x^{2} - 35 \, b^{3} c^{2} d^{5} x - b^{4} c d^{5}\right )} e^{5} + {\left (1280 \, c^{5} d^{6} x^{3} - 2560 \, b c^{4} d^{6} x^{2} + 675 \, b^{2} c^{3} d^{6} x - 7 \, b^{3} c^{2} d^{6}\right )} e^{4} + 5 \, {\left (256 \, c^{5} d^{7} x^{2} - 256 \, b c^{4} d^{7} x + 27 \, b^{2} c^{3} d^{7}\right )} e^{3} + 128 \, {\left (5 \, c^{5} d^{8} x - 2 \, b c^{4} d^{8}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{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 )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{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 )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (128 \, c^{5} d^{8} e^{2} - 2 \, b^{4} c x^{4} e^{10} - {\left (14 \, b^{3} c^{2} d x^{4} + 9 \, b^{4} c d x^{3}\right )} e^{9} + {\left (207 \, b^{2} c^{3} d^{2} x^{4} + 8 \, b^{3} c^{2} d^{2} x^{3}\right )} e^{8} - 2 \, {\left (193 \, b c^{4} d^{3} x^{4} - 291 \, b^{2} c^{3} d^{3} x^{3} + 5 \, b^{3} c^{2} d^{3} x^{2}\right )} e^{7} + {\left (193 \, c^{5} d^{4} x^{4} - 1239 \, b c^{4} d^{4} x^{3} + 783 \, b^{2} c^{3} d^{4} x^{2} - 5 \, b^{3} c^{2} d^{4} x\right )} e^{6} + {\left (650 \, c^{5} d^{5} x^{3} - 1665 \, b c^{4} d^{5} x^{2} + 477 \, b^{2} c^{3} d^{5} x - b^{3} c^{2} d^{5}\right )} e^{5} + {\left (880 \, c^{5} d^{6} x^{2} - 1024 \, b c^{4} d^{6} x + 111 \, b^{2} c^{3} d^{6}\right )} e^{4} + 16 \, {\left (34 \, c^{5} d^{7} x - 15 \, b c^{4} d^{7}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{189 \, {\left (c^{3} d^{9} e^{7} + b^{2} c d^{2} x^{5} e^{14} - {\left (2 \, b c^{2} d^{3} x^{5} - 5 \, b^{2} c d^{3} x^{4}\right )} e^{13} + {\left (c^{3} d^{4} x^{5} - 10 \, b c^{2} d^{4} x^{4} + 10 \, b^{2} c d^{4} x^{3}\right )} e^{12} + 5 \, {\left (c^{3} d^{5} x^{4} - 4 \, b c^{2} d^{5} x^{3} + 2 \, b^{2} c d^{5} x^{2}\right )} e^{11} + 5 \, {\left (2 \, c^{3} d^{6} x^{3} - 4 \, b c^{2} d^{6} x^{2} + b^{2} c d^{6} x\right )} e^{10} + {\left (10 \, c^{3} d^{7} x^{2} - 10 \, b c^{2} d^{7} x + b^{2} c d^{7}\right )} e^{9} + {\left (5 \, c^{3} d^{8} x - 2 \, b c^{2} d^{8}\right )} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 - 2*b^5*x^5*e^10 - (13*b^4*c*d*x^5 + 10*b^5*d*x^4)*e^9 - (77*b^3*c^2*d^2*x^5 + 65*b^4*c*
d^2*x^4 + 20*b^5*d^2*x^3)*e^8 + (478*b^2*c^3*d^3*x^5 - 385*b^3*c^2*d^3*x^4 - 130*b^4*c*d^3*x^3 - 20*b^5*d^3*x^
2)*e^7 - 10*(64*b*c^4*d^4*x^5 - 239*b^2*c^3*d^4*x^4 + 77*b^3*c^2*d^4*x^3 + 13*b^4*c*d^4*x^2 + b^5*d^4*x)*e^6 +
(256*c^5*d^5*x^5 - 3200*b*c^4*d^5*x^4 + 4780*b^2*c^3*d^5*x^3 - 770*b^3*c^2*d^5*x^2 - 65*b^4*c*d^5*x - 2*b^5*d
^5)*e^5 + (1280*c^5*d^6*x^4 - 6400*b*c^4*d^6*x^3 + 4780*b^2*c^3*d^6*x^2 - 385*b^3*c^2*d^6*x - 13*b^4*c*d^6)*e^
4 + (2560*c^5*d^7*x^3 - 6400*b*c^4*d^7*x^2 + 2390*b^2*c^3*d^7*x - 77*b^3*c^2*d^7)*e^3 + 2*(1280*c^5*d^8*x^2 -
1600*b*c^4*d^8*x + 239*b^2*c^3*d^8)*e^2 + 640*(2*c^5*d^9*x - b*c^4*d^9)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse
(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*e
^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) + 6*(128*c^5*d^9*e - b^4*c*x^5*e^10 - (7*b^3*c^2*d*x^5 + 5*b^4*
c*d*x^4)*e^9 + 5*(27*b^2*c^3*d^2*x^5 - 7*b^3*c^2*d^2*x^4 - 2*b^4*c*d^2*x^3)*e^8 - (256*b*c^4*d^3*x^5 - 675*b^2
*c^3*d^3*x^4 + 70*b^3*c^2*d^3*x^3 + 10*b^4*c*d^3*x^2)*e^7 + (128*c^5*d^4*x^5 - 1280*b*c^4*d^4*x^4 + 1350*b^2*c
^3*d^4*x^3 - 70*b^3*c^2*d^4*x^2 - 5*b^4*c*d^4*x)*e^6 + (640*c^5*d^5*x^4 - 2560*b*c^4*d^5*x^3 + 1350*b^2*c^3*d^
5*x^2 - 35*b^3*c^2*d^5*x - b^4*c*d^5)*e^5 + (1280*c^5*d^6*x^3 - 2560*b*c^4*d^6*x^2 + 675*b^2*c^3*d^6*x - 7*b^3
*c^2*d^6)*e^4 + 5*(256*c^5*d^7*x^2 - 256*b*c^4*d^7*x + 27*b^2*c^3*d^7)*e^3 + 128*(5*c^5*d^8*x - 2*b*c^4*d^8)*e
^2)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^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)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c
)) + 3*(128*c^5*d^8*e^2 - 2*b^4*c*x^4*e^10 - (14*b^3*c^2*d*x^4 + 9*b^4*c*d*x^3)*e^9 + (207*b^2*c^3*d^2*x^4 + 8
*b^3*c^2*d^2*x^3)*e^8 - 2*(193*b*c^4*d^3*x^4 - 291*b^2*c^3*d^3*x^3 + 5*b^3*c^2*d^3*x^2)*e^7 + (193*c^5*d^4*x^4
- 1239*b*c^4*d^4*x^3 + 783*b^2*c^3*d^4*x^2 - 5*b^3*c^2*d^4*x)*e^6 + (650*c^5*d^5*x^3 - 1665*b*c^4*d^5*x^2 + 4
77*b^2*c^3*d^5*x - b^3*c^2*d^5)*e^5 + (880*c^5*d^6*x^2 - 1024*b*c^4*d^6*x + 111*b^2*c^3*d^6)*e^4 + 16*(34*c^5*
d^7*x - 15*b*c^4*d^7)*e^3)*sqrt(c*x^2 + b*x)*sqrt(x*e + d))/(c^3*d^9*e^7 + b^2*c*d^2*x^5*e^14 - (2*b*c^2*d^3*x
^5 - 5*b^2*c*d^3*x^4)*e^13 + (c^3*d^4*x^5 - 10*b*c^2*d^4*x^4 + 10*b^2*c*d^4*x^3)*e^12 + 5*(c^3*d^5*x^4 - 4*b*c
^2*d^5*x^3 + 2*b^2*c*d^5*x^2)*e^11 + 5*(2*c^3*d^6*x^3 - 4*b*c^2*d^6*x^2 + b^2*c*d^6*x)*e^10 + (10*c^3*d^7*x^2
- 10*b*c^2*d^7*x + b^2*c*d^7)*e^9 + (5*c^3*d^8*x - 2*b*c^2*d^8)*e^8)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/(x*e + d)^(11/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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