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

Optimal. Leaf size=449 \[ -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[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)*(c*x/b+1)^(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)*(c*x/b+1)^(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

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Rubi [A]
time = 0.39, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {826, 828, 857, 729, 113, 111, 118, 117} \begin {gather*} -\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 ) F\left (\text {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 (\text {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}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains complex when optimal does not.
time = 22.47, size = 514, normalized size = 1.14 \begin {gather*} \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 \sinh ^{-1}\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} F\left (i \sinh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1609\) vs. \(2(395)=790\).
time = 0.62, size = 1610, normalized size = 3.59

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 \left (c e \,x^{2}+b e x \right ) d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 B c \,x^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{7 e^{2}}+\frac {2 \left (\frac {c \left (A c e +2 b B e -B c d \right )}{e^{2}}-\frac {2 B c \left (3 b e +3 c d \right )}{7 e^{2}}\right ) x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{5 c e}+\frac {2 \left (\frac {2 A b c \,e^{2}-A \,c^{2} d e +B \,e^{2} b^{2}-2 e B b c d +B \,c^{2} d^{2}}{e^{3}}-\frac {5 B c b d}{7 e^{2}}-\frac {2 \left (\frac {c \left (A c e +2 b B e -B c d \right )}{e^{2}}-\frac {2 B c \left (3 b e +3 c d \right )}{7 e^{2}}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 c e}+\frac {2 \left (-\frac {d \left (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}\right )}{e^{5}}+\frac {d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right ) \left (b e -c d \right )}{e^{5}}-\frac {b d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{e^{4}}-\frac {\left (\frac {2 A b c \,e^{2}-A \,c^{2} d e +B \,e^{2} b^{2}-2 e B b c d +B \,c^{2} d^{2}}{e^{3}}-\frac {5 B c b d}{7 e^{2}}-\frac {2 \left (\frac {c \left (A c e +2 b B e -B c d \right )}{e^{2}}-\frac {2 B c \left (3 b e +3 c d \right )}{7 e^{2}}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) b d}{3 c e}\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 {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^{4}}-\frac {d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right ) c}{e^{4}}-\frac {3 \left (\frac {c \left (A c e +2 b B e -B c d \right )}{e^{2}}-\frac {2 B c \left (3 b e +3 c d \right )}{7 e^{2}}\right ) b d}{5 c e}-\frac {2 \left (\frac {2 A b c \,e^{2}-A \,c^{2} d e +B \,e^{2} b^{2}-2 e B b c d +B \,c^{2} d^{2}}{e^{3}}-\frac {5 B c b d}{7 e^{2}}-\frac {2 \left (\frac {c \left (A c e +2 b B e -B c d \right )}{e^{2}}-\frac {2 B c \left (3 b e +3 c d \right )}{7 e^{2}}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \left (b e +c d \right )}{3 c e}\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 )}\)	\(1152\)
default	\(\text {Expression too large to display}\)	\(1610\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-9*B*b^2*c^3*e^4*x^3-16*B*c^5*d^2*e^2*x^3-14*A*b^2*c^3*e^4*x^2+56*A*c^5
*d^2*e^2*x^2-B*b^3*c^2*e^4*x^2-64*B*c^5*d^3*e*x^2-119*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e^3+224*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2*e^2-112*A*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c
^4*d^3*e-B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*b^4*c*d*e^3-71*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b
)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^2+200*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e-9*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^3+7*A*((c*x+b)/b)^(1
/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*e^4-1
28*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*b*c^4*d^4+128*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^4-13*B*b*c^4*e^4*x^4+8*B*c^5*d*e^3*x^4-21*A*b*c^4*e^4*x^3+14*A*c^5*d*e^3*x^
3-5*B*c^5*e^4*x^5-7*A*c^5*e^4*x^4+25*B*b*c^4*d*e^3*x^3-168*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2*e^2+112*A*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^3*e+56*A*((c*
x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b
^3*c^2*d*e^3-264*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e+44*B*b*c^4*d^2*e^2*x^2-35*A*b*c^4*d*e^3*x^2+147*B*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^2+16*B*
b^2*c^3*d*e^3*x^2+60*B*b^2*c^3*d^2*e^2*x-64*B*b*c^4*d^3*e*x-2*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)
*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^4-49*A*b^2*c^3*d*e^3*x+56*A*b*c^4*d^2
*e^2*x-B*b^3*c^2*d*e^3*x)/c^3/e^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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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((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)/(x*e + d)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.10, size = 811, normalized size = 1.81 \begin {gather*} \frac {2 \, {\left ({\left (128 \, B c^{4} d^{5} + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} x e^{5} + {\left (2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d x + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} d\right )} e^{4} + {\left ({\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{2} x + 2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d^{2}\right )} e^{3} - {\left (8 \, {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3} x - {\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{3}\right )} e^{2} + 8 \, {\left (16 \, B c^{4} d^{4} x - {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{4}\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 ) + 3 \, {\left (128 \, B c^{4} d^{4} e + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x e^{5} + {\left ({\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d x + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} d\right )} e^{4} - {\left (8 \, {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2} x - {\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d^{2}\right )} e^{3} + 8 \, {\left (16 \, B c^{4} d^{3} x - {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3}\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 (64 \, B c^{4} d^{3} e^{2} + {\left (5 \, B c^{4} x^{3} + {\left (8 \, B b c^{3} + 7 \, A c^{4}\right )} x^{2} + {\left (B b^{2} c^{2} + 14 \, A b c^{3}\right )} x\right )} e^{5} - {\left (8 \, B c^{4} d x^{2} + {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d x - {\left (B b^{2} c^{2} + 49 \, A b c^{3}\right )} d\right )} e^{4} + 4 \, {\left (4 \, B c^{4} d^{2} x - {\left (15 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{105 \, {\left (c^{3} x e^{7} + c^{3} d e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + (2*B*b^4 - 7*A*b^3*c)*x*e^5 + (2*(5*B*b^3*c - 21*A*b^2*c^2)*d*x + (2*B*b^4 - 7*A*b^3*c
)*d)*e^4 + ((55*B*b^2*c^2 + 168*A*b*c^3)*d^2*x + 2*(5*B*b^3*c - 21*A*b^2*c^2)*d^2)*e^3 - (8*(25*B*b*c^3 + 14*A
*c^4)*d^3*x - (55*B*b^2*c^2 + 168*A*b*c^3)*d^3)*e^2 + 8*(16*B*c^4*d^4*x - (25*B*b*c^3 + 14*A*c^4)*d^4)*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) + 3*(128*B*c^4*d^4*e + (2*B*b^3*
c - 7*A*b^2*c^2)*x*e^5 + ((11*B*b^2*c^2 + 112*A*b*c^3)*d*x + (2*B*b^3*c - 7*A*b^2*c^2)*d)*e^4 - (8*(17*B*b*c^3
+ 14*A*c^4)*d^2*x - (11*B*b^2*c^2 + 112*A*b*c^3)*d^2)*e^3 + 8*(16*B*c^4*d^3*x - (17*B*b*c^3 + 14*A*c^4)*d^3)*
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*(64*B*c^4*d^3*e^2 + (5*B*c^4*x^3 + (8*B*b*c^3 + 7*A*c^4)*x^2 + (B*b^2*c^2 + 14*A*b*c^3)*x)*e^5 - (8*B*
c^4*d*x^2 + (17*B*b*c^3 + 14*A*c^4)*d*x - (B*b^2*c^2 + 49*A*b*c^3)*d)*e^4 + 4*(4*B*c^4*d^2*x - (15*B*b*c^3 + 1
4*A*c^4)*d^2)*e^3)*sqrt(c*x^2 + b*x)*sqrt(x*e + d))/(c^3*x*e^7 + c^3*d*e^6)

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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 {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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((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)/(x*e + d)^(3/2), 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 )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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