\(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\) [404]

Optimal result

Integrand size = 23, antiderivative size = 474 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 d e^6 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/21*(c*d^2*(-13*b*e+16*c*d)+e*(3*b^2*e^2-22*b*c*d*e+22*c^2*d^2)*x)*(c*x^2+b*x)^(3/2)/d/e^3/(-b*e+c*d)/(e*x+d
)^(5/2)-2/7*(c*x^2+b*x)^(5/2)/e/(e*x+d)^(7/2)-2/21*(-b*e+2*c*d)*(3*b^2*e^2-128*b*c*d*e+128*c^2*d^2)*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/e^6/(-b
*e+c*d)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+4/21*(27*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)/e^6/(e*x+d)^(1/2)/(c*x^
2+b*x)^(1/2)+2/21*c*(d*(51*b^2*e^2-176*b*c*d*e+128*c^2*d^2)+e*(3*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*x)*(c*x^2+b*x)
^(1/2)/d/e^5/(-b*e+c*d)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {746, 824, 826, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{21 e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 d e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 c \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{21 d e^5 \sqrt {d+e x} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{21 d e^3 (d+e x)^{5/2} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(2*c*(d*(128*c^2*d^2 - 176*b*c*d*e + 51*b^2*e^2) + e*(32*c^2*d^2 - 32*b*c*d*e + 3*b^2*e^2)*x)*Sqrt[b*x + c*x^2
])/(21*d*e^5*(c*d - b*e)*Sqrt[d + e*x]) - (2*(c*d^2*(16*c*d - 13*b*e) + e*(22*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2
)*x)*(b*x + c*x^2)^(3/2))/(21*d*e^3*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(b*x + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2
)) - (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e + 3*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)])/(21*d*e^6*(c*d - b*e)*Sqrt[1 + (e*x)/d]*S
qrt[b*x + c*x^2]) + (4*Sqrt[-b]*Sqrt[c]*(128*c^2*d^2 - 128*b*c*d*e + 27*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(21*e^6*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 826

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

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}}{7 e (d+e x)^{7/2}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e} \\ & = -\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \int \frac {\left (-\frac {1}{2} b c d (16 c d-13 b e)-\frac {1}{2} c \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx}{7 d e^3 (c d-b e)} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 d e^5 (c d-b e)} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{21 d e^6 (c d-b e)}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 e^6} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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}{21 d e^6 (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+27 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}{21 e^6 \sqrt {b x+c x^2}} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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}{21 d e^6 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+27 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}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 c \left (d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )+e \left (32 c^2 d^2-32 b c d e+3 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{21 d e^5 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (c d^2 (16 c d-13 b e)+e \left (22 c^2 d^2-22 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{21 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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 )}{21 d e^6 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+27 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 )}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.58 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.05 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (3 b^3 e^6 x^3-b^2 c d e^2 \left (51 d^3+169 d^2 e x+194 d e^2 x^2+85 e^3 x^3\right )-c^3 d^2 \left (128 d^4+416 d^3 e x+464 d^2 e^2 x^2+186 d e^3 x^3+7 e^4 x^4\right )+b c^2 d e \left (176 d^4+576 d^3 e x+649 d^2 e^2 x^2+265 d e^3 x^3+7 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} c (d+e x)^3 \left (\sqrt {\frac {b}{c}} \left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) (b+c x) (d+e x)+i b e \left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (128 c^3 d^3-208 b c^2 d^2 e+83 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{21 b d e^6 (c d-b e) x^3 (b+c x)^3 (d+e x)^{7/2}} \]

[In]

Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(3*b^3*e^6*x^3 - b^2*c*d*e^2*(51*d^3 + 169*d^2*e*x + 194*d*e^2*x^2 +
85*e^3*x^3) - c^3*d^2*(128*d^4 + 416*d^3*e*x + 464*d^2*e^2*x^2 + 186*d*e^3*x^3 + 7*e^4*x^4) + b*c^2*d*e*(176*d
^4 + 576*d^3*e*x + 649*d^2*e^2*x^2 + 265*d*e^3*x^3 + 7*e^4*x^4)) + Sqrt[b/c]*c*(d + e*x)^3*(Sqrt[b/c]*(256*c^3
*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(256*c^3*d^3 - 384*b*c^2*d^2
*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sq
rt[x]], (c*d)/(b*e)] - I*b*e*(128*c^3*d^3 - 208*b*c^2*d^2*e + 83*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sq
rt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(21*b*d*e^6*(c*d - b*e)*x^3*(b
+ c*x)^3*(d + e*x)^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(414)=828\).

Time = 3.04 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.89

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}}{7 e^{9} \left (x +\frac {d}{e}\right )^{4}}+\frac {6 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}}{7 e^{8} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (9 b^{2} e^{2}-52 b c d e +52 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{21 e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}\right )}{21 d \left (b e -c d \right ) e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{5}}+\frac {2 \left (\frac {c \left (3 b^{2} e^{2}-12 b c d e +10 c^{2} d^{2}\right )}{e^{6}}-\frac {c \left (9 b^{2} e^{2}-52 b c d e +52 c^{2} d^{2}\right )}{21 e^{6}}+\frac {3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}}{21 e^{6} d}-\frac {b \left (3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}\right )}{21 e^{5} d \left (b e -c d \right )}-\frac {c^{2} b d}{3 e^{5}}\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^{2} \left (3 b e -4 c d \right )}{e^{5}}-\frac {c \left (3 b^{3} e^{3}-85 b^{2} d \,e^{2} c +237 b \,c^{2} d^{2} e -158 c^{3} d^{3}\right )}{21 e^{5} d \left (b e -c d \right )}-\frac {2 c^{2} \left (b e +c d \right )}{3 e^{5}}\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 )}\)	\(894\)
default	\(\text {Expression too large to display}\)	\(3284\)

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),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)*(-2/7*d^2*(b^2*e^2-2*b*c*d*e+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+6/7*d*(b^2*e^2-3*b*c*d*e+2*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)^3-2/21*(9*b^2*e^2-52*b*c*d*e+52*c^2*d^2)/e^7*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(
x+d/e)^2+2/21*(c*e*x^2+b*e*x)/d/(b*e-c*d)/e^6*(3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)/((x+d/e)*
(c*e*x^2+b*e*x))^(1/2)+2/3*c^2/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(c*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2
)/e^6-1/21*c*(9*b^2*e^2-52*b*c*d*e+52*c^2*d^2)/e^6+1/21/e^6*(3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*
d^3)/d-1/21*b/e^5/d/(b*e-c*d)*(3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)-1/3*c^2/e^5*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*(c^2/e^5*(3*b*e-4*c*d)-1/21/e^5*c*(3*b^3*e^3-85*b^2*c*d*e
^2+237*b*c^2*d^2*e-158*c^3*d^3)/d/(b*e-c*d)-2/3*c^2/e^5*(b*e+c*d))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+
d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-1/c*b+d/e)*EllipticE(((1/c*b+x)*c/b)^(1/2)
,(-1/c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.37 (sec) , antiderivative size = 1184, normalized size of antiderivative = 2.50 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

2/63*((256*c^4*d^8 - 512*b*c^3*d^7*e + 278*b^2*c^2*d^6*e^2 - 22*b^3*c*d^5*e^3 - 3*b^4*d^4*e^4 + (256*c^4*d^4*e
^4 - 512*b*c^3*d^3*e^5 + 278*b^2*c^2*d^2*e^6 - 22*b^3*c*d*e^7 - 3*b^4*e^8)*x^4 + 4*(256*c^4*d^5*e^3 - 512*b*c^
3*d^4*e^4 + 278*b^2*c^2*d^3*e^5 - 22*b^3*c*d^2*e^6 - 3*b^4*d*e^7)*x^3 + 6*(256*c^4*d^6*e^2 - 512*b*c^3*d^5*e^3
+ 278*b^2*c^2*d^4*e^4 - 22*b^3*c*d^3*e^5 - 3*b^4*d^2*e^6)*x^2 + 4*(256*c^4*d^7*e - 512*b*c^3*d^6*e^2 + 278*b^
2*c^2*d^5*e^3 - 22*b^3*c*d^4*e^4 - 3*b^4*d^3*e^5)*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*(256*c^4*d^7*e - 384*b*c^3*d^6*e^2 + 134*b^2*c^2*d^5*e^3 - 3*b^3*c*d^4*e^4 + (256*c^4*d^3*e^
5 - 384*b*c^3*d^2*e^6 + 134*b^2*c^2*d*e^7 - 3*b^3*c*e^8)*x^4 + 4*(256*c^4*d^4*e^4 - 384*b*c^3*d^3*e^5 + 134*b^
2*c^2*d^2*e^6 - 3*b^3*c*d*e^7)*x^3 + 6*(256*c^4*d^5*e^3 - 384*b*c^3*d^4*e^4 + 134*b^2*c^2*d^3*e^5 - 3*b^3*c*d^
2*e^6)*x^2 + 4*(256*c^4*d^6*e^2 - 384*b*c^3*d^5*e^3 + 134*b^2*c^2*d^4*e^4 - 3*b^3*c*d^3*e^5)*x)*sqrt(c*e)*weie
rstrassZeta(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^4*d^6*e^2 - 176*b
*c^3*d^5*e^3 + 51*b^2*c^2*d^4*e^4 + 7*(c^4*d^2*e^6 - b*c^3*d*e^7)*x^4 + (186*c^4*d^3*e^5 - 265*b*c^3*d^2*e^6 +
85*b^2*c^2*d*e^7 - 3*b^3*c*e^8)*x^3 + (464*c^4*d^4*e^4 - 649*b*c^3*d^3*e^5 + 194*b^2*c^2*d^2*e^6)*x^2 + (416*
c^4*d^5*e^3 - 576*b*c^3*d^4*e^4 + 169*b^2*c^2*d^3*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^2*d^6*e^7 - b*c*
d^5*e^8 + (c^2*d^2*e^11 - b*c*d*e^12)*x^4 + 4*(c^2*d^3*e^10 - b*c*d^2*e^11)*x^3 + 6*(c^2*d^4*e^9 - b*c*d^3*e^1
0)*x^2 + 4*(c^2*d^5*e^8 - b*c*d^4*e^9)*x)

Sympy [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]

[In]

integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(9/2), x)

Maxima [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(9/2), x)

Giac [F]

\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

[In]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x)

[Out]

int((b*x + c*x^2)^(5/2)/(d + e*x)^(9/2), x)