\(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [1261]

Optimal result

Integrand size = 28, antiderivative size = 449 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

2/7*(B*e*x-7*A*e+8*B*d)*(c*x^2+b*x)^(3/2)/e^2/(e*x+d)^(1/2)+2/35*(5*b*c*e*(-7*A*e+8*B*d)*(-b*e+2*c*d)-(-14*A*c
*e-B*b*e+16*B*c*d)*(-2*b^2*e^2-3*b*c*d*e+8*c^2*d^2))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b
)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/c^(3/2)/e^5/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/35*d*(-b*e+c*d)*
(56*A*c*e*(-b*e+2*c*d)-B*(-b^2*e^2-72*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)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(3/2)/e^5/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/35*(7*A*c
*e*(-7*b*e+8*c*d)-B*(b^2*e^2-60*b*c*d*e+64*c^2*d^2)+3*c*e*(-14*A*c*e-B*b*e+16*B*c*d)*x)*(e*x+d)^(1/2)*(c*x^2+b
*x)^(1/2)/c/e^4

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {826, 828, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (56 A c e (2 c d-b e)-B \left (-b^2 e^2-72 b c d e+128 c^2 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (-14 A c e-b B e+16 B c d)\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (3 c e x (-14 A c e-b B e+16 B c d)+7 A c e (8 c d-7 b e)-B \left (b^2 e^2-60 b c d e+64 c^2 d^2\right )\right )}{35 c e^4}+\frac {2 \left (b x+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}} \]

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(7*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 60*b*c*d*e + b^2*e^2) + 3*c*e*(16*B*c*d - b*B*e -
14*A*c*e)*x)*Sqrt[b*x + c*x^2])/(35*c*e^4) + (2*(8*B*d - 7*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d +
e*x]) + (2*Sqrt[-b]*(5*b*c*e*(8*B*d - 7*A*e)*(2*c*d - b*e) - (16*B*c*d - b*B*e - 14*A*c*e)*(8*c^2*d^2 - 3*b*c*
d*e - 2*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)])/(35*c^(3/2)*e^5*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(56*A*c*e*(2*c*d - b*
e) - B*(128*c^2*d^2 - 72*b*c*d*e - b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*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 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 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {6 \int \frac {\left (\frac {1}{2} b (8 B d-7 A e)+\frac {1}{2} (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^2} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} b d \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )\right )+\frac {1}{4} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-2 (16 B c d-b B e-14 A c e) \left (4 c^2 d^2-\frac {3}{2} b c d e-b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{35 c e^4} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{35 c e^5}-\frac {\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{35 c e^5} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{35 c e^5 \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{35 c e^5 \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\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}{35 c e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\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}{35 c e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\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 )}{35 c^{3/2} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\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 )}{35 c^{3/2} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.52 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (35 c d (B d-A e) (c d-b e)+\left (7 A c e (-3 c d+2 b e)+B \left (29 c^2 d^2-25 b c d e+b^2 e^2\right )\right ) (d+e x)+c e (-13 B c d+8 b B e+7 A c e) x (d+e x)+5 B c^2 e^2 x^2 (d+e x)\right )+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (7 A c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B \left (128 c^3 d^3-136 b c^2 d^2 e+11 b^2 c d e^2+2 b^3 e^3\right )\right ) (b+c x) (d+e x)+i b e \left (7 A c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B \left (128 c^3 d^3-136 b c^2 d^2 e+11 b^2 c d e^2+2 b^3 e^3\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (7 A c e (8 c d-b e)+2 B \left (-32 c^2 d^2+6 b c d e+b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{35 b c e^5 x^2 (b+c x)^2 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(35*c*d*(B*d - A*e)*(c*d - b*e) + (7*A*c*e*(-3*c*d + 2*b*e) + B*(29*c^
2*d^2 - 25*b*c*d*e + b^2*e^2))*(d + e*x) + c*e*(-13*B*c*d + 8*b*B*e + 7*A*c*e)*x*(d + e*x) + 5*B*c^2*e^2*x^2*(
d + e*x)) + Sqrt[b/c]*(Sqrt[b/c]*(7*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*(128*c^3*d^3 - 136*b*c^2*d^2
*e + 11*b^2*c*d*e^2 + 2*b^3*e^3))*(b + c*x)*(d + e*x) + I*b*e*(7*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B
*(128*c^3*d^3 - 136*b*c^2*d^2*e + 11*b^2*c*d*e^2 + 2*b^3*e^3))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ell
ipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(7*A*c*e*(8*c*d - b*e) + 2*B*(-32*c^2*d^
2 + 6*b*c*d*e + b^2*e^2))*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)])))/(35*b*c*e^5*x^2*(b + c*x)^2*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1151\) vs. \(2(395)=790\).

Time = 0.89 (sec) , antiderivative size = 1152, normalized size of antiderivative = 2.57

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1152\)
default	\(\text {Expression too large to display}\)	\(1610\)

[In]

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((e*x+d)*x*(c*x+b))^(1/2)/x/(c*x+b)*(2*(c*e*x^2+b*e*x)*d*(A*b*e^2-A*c*d*e-B*
b*d*e+B*c*d^2)/e^5/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/7*B*c/e^2*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/5*(
c/e^2*(A*c*e+2*B*b*e-B*c*d)-2/7*B*c/e^2*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(1/e^3*
(2*A*b*c*e^2-A*c^2*d*e+B*b^2*e^2-2*B*b*c*d*e+B*c^2*d^2)-5/7*B*c/e^2*b*d-2/5*(c/e^2*(A*c*e+2*B*b*e-B*c*d)-2/7*B
*c/e^2*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(-d*(A*b^2*e^3-2*A*b*c*d*
e^2+A*c^2*d^2*e-B*b^2*d*e^2+2*B*b*c*d^2*e-B*c^2*d^3)/e^5+d*(A*b*e^2-A*c*d*e-B*b*d*e+B*c*d^2)/e^5*(b*e-c*d)-b/e
^4*d*(A*b*e^2-A*c*d*e-B*b*d*e+B*c*d^2)-1/3*(1/e^3*(2*A*b*c*e^2-A*c^2*d*e+B*b^2*e^2-2*B*b*c*d*e+B*c^2*d^2)-5/7*
B*c/e^2*b*d-2/5*(c/e^2*(A*c*e+2*B*b*e-B*c*d)-2/7*B*c/e^2*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*b*d)*b/c*((x+b/
c)/b*c)^(1/2)*((x+d/e)/(-b/c+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(((x+b/
c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(1/e^4*(A*b^2*e^3-2*A*b*c*d*e^2+A*c^2*d^2*e-B*b^2*d*e^2+2*B*b*c*d^2*e
-B*c^2*d^3)-d*(A*b*e^2-A*c*d*e-B*b*d*e+B*c*d^2)/e^4*c-3/5*(c/e^2*(A*c*e+2*B*b*e-B*c*d)-2/7*B*c/e^2*(3*b*e+3*c*
d))/c/e*b*d-2/3*(1/e^3*(2*A*b*c*e^2-A*c^2*d*e+B*b^2*e^2-2*B*b*c*d*e+B*c^2*d^2)-5/7*B*c/e^2*b*d-2/5*(c/e^2*(A*c
*e+2*B*b*e-B*c*d)-2/7*B*c/e^2*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(b*e+c*d))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e
)/(-b/c+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)*((-b/c+d/e)*EllipticE(((x+b/c)/b*c)^(
1/2),(-b/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.17 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.86 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (128 \, B c^{4} d^{5} - 8 \, {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{4} e + {\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{3} e^{2} + 2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d^{2} e^{3} + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} d e^{4} + {\left (128 \, B c^{4} d^{4} e - 8 \, {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3} e^{2} + {\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{2} e^{3} + 2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d e^{4} + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} e^{5}\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 (128 \, B c^{4} d^{4} e - 8 \, {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3} e^{2} + {\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d^{2} e^{3} + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} d e^{4} + {\left (128 \, B c^{4} d^{3} e^{2} - 8 \, {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2} e^{3} + {\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d e^{4} + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} e^{5}\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 (5 \, B c^{4} e^{5} x^{3} + 64 \, B c^{4} d^{3} e^{2} - 4 \, {\left (15 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2} e^{3} + {\left (B b^{2} c^{2} + 49 \, A b c^{3}\right )} d e^{4} - {\left (8 \, B c^{4} d e^{4} - {\left (8 \, B b c^{3} + 7 \, A c^{4}\right )} e^{5}\right )} x^{2} + {\left (16 \, B c^{4} d^{2} e^{3} - {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d e^{4} + {\left (B b^{2} c^{2} + 14 \, A b c^{3}\right )} e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{105 \, {\left (c^{3} e^{7} x + c^{3} d e^{6}\right )}} \]

[In]

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

[Out]

2/105*((128*B*c^4*d^5 - 8*(25*B*b*c^3 + 14*A*c^4)*d^4*e + (55*B*b^2*c^2 + 168*A*b*c^3)*d^3*e^2 + 2*(5*B*b^3*c
- 21*A*b^2*c^2)*d^2*e^3 + (2*B*b^4 - 7*A*b^3*c)*d*e^4 + (128*B*c^4*d^4*e - 8*(25*B*b*c^3 + 14*A*c^4)*d^3*e^2 +
(55*B*b^2*c^2 + 168*A*b*c^3)*d^2*e^3 + 2*(5*B*b^3*c - 21*A*b^2*c^2)*d*e^4 + (2*B*b^4 - 7*A*b^3*c)*e^5)*x)*sqr
t(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*(128*B*c^4*d^4*e - 8*(17*B*b*c^3 + 14*
A*c^4)*d^3*e^2 + (11*B*b^2*c^2 + 112*A*b*c^3)*d^2*e^3 + (2*B*b^3*c - 7*A*b^2*c^2)*d*e^4 + (128*B*c^4*d^3*e^2 -
8*(17*B*b*c^3 + 14*A*c^4)*d^2*e^3 + (11*B*b^2*c^2 + 112*A*b*c^3)*d*e^4 + (2*B*b^3*c - 7*A*b^2*c^2)*e^5)*x)*sq
rt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*
c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*
d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(5*B*c^4*e^5
*x^3 + 64*B*c^4*d^3*e^2 - 4*(15*B*b*c^3 + 14*A*c^4)*d^2*e^3 + (B*b^2*c^2 + 49*A*b*c^3)*d*e^4 - (8*B*c^4*d*e^4
- (8*B*b*c^3 + 7*A*c^4)*e^5)*x^2 + (16*B*c^4*d^2*e^3 - (17*B*b*c^3 + 14*A*c^4)*d*e^4 + (B*b^2*c^2 + 14*A*b*c^3
)*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^3*e^7*x + c^3*d*e^6)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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