3.2474 \(\int \frac{1}{(d+e x)^{3/2} (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=607 \[ -\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{4 e \sqrt{a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a
+ b*x + c*x^2]) - (4*e*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2)^2*Sqrt[d + e*x]) + (2*Sqrt[2]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*
(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sq
rt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e
^2)^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*(2*c*d - b*e
)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti
cF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.528559, antiderivative size = 607, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {740, 834, 843, 718, 424, 419} \[ -\frac{4 e \sqrt{a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a
+ b*x + c*x^2]) - (4*e*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2)^2*Sqrt[d + e*x]) + (2*Sqrt[2]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*
(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sq
rt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e
^2)^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*(2*c*d - b*e
)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti
cF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 843

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]

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, 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] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (b c d-2 b^2 e+6 a c e\right )+\frac{1}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}+\frac{4 \int \frac{\frac{1}{4} c e \left (b c d^2+b^2 d e-8 a c d e+a b e^2\right )+\frac{1}{2} c e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{(c (2 c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}+\frac{\left (2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}+\frac{\left (2 \sqrt{2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}-\frac{\left (2 \sqrt{2} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}+\frac{2 \sqrt{2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 10.9627, size = 784, normalized size = 1.29 \[ \frac{-\frac{i (d+e x) \sqrt{1-\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{4 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+2} \left (\left (c \left (a e^2 \left (3 \sqrt{e^2 \left (b^2-4 a c\right )}+8 c d\right )-c d^2 \sqrt{e^2 \left (b^2-4 a c\right )}\right )-b^2 e^2 \left (\sqrt{e^2 \left (b^2-4 a c\right )}+2 c d\right )+b c e \left (d \sqrt{e^2 \left (b^2-4 a c\right )}-4 a e^2\right )+b^3 e^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a e^2-b d e+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}{\sqrt{d+e x}}\right ),-\frac{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}{\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d}\right )+\left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{e \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}-\frac{4 e (a+x (b+c x)) \left (c e (3 a e+b d)-b^2 e^2-c^2 d^2\right )}{\sqrt{d+e x}}-\frac{2 \left ((d+e x) \left (b c \left (c d (d-2 e x)-3 a e^2\right )+2 c^2 \left (a e (2 d-e x)+c d^2 x\right )+b^2 c e (e x-2 d)+b^3 e^2\right )+e^3 \left (b^2-4 a c\right ) (a+x (b+c x))\right )}{\sqrt{d+e x}}}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((-4*e*(-(c^2*d^2) - b^2*e^2 + c*e*(b*d + 3*a*e))*(a + x*(b + c*x)))/Sqrt[d + e*x] - (2*((b^2 - 4*a*c)*e^3*(a
+ x*(b + c*x)) + (d + e*x)*(b^3*e^2 + b^2*c*e*(-2*d + e*x) + b*c*(-3*a*e^2 + c*d*(d - 2*e*x)) + 2*c^2*(c*d^2*x
+ a*e*(2*d - e*x)))))/Sqrt[d + e*x] - (I*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sq
rt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)
*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*EllipticE[
I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((
-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (b^3*e^3 - b^2*e^2*(2*c*d
+ Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]
) + a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*
c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e
+ Sqrt[(b^2 - 4*a*c)*e^2]))]))/(e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]))
/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.386, size = 4415, normalized size = 7.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x)

[Out]

(3*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(
-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^
(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*
a*c+b^2)^(1/2)))^(1/2))*b^3*d*e^3+12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+
(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1
/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^
(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*c*e^4+4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2
)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a
*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e
-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*c^3*d^4-6*a*b*c*d*e
^3-2*x*b*c^2*d^2*e^2+4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(
1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)
)^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*
d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*e^4-4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^
(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))
/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)
,(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^3*d*e^3-3*2^(1/2)*(-(e*x+d)*c/(
e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2
)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4
*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*
a*b^2*e^4-4*b^2*c*d^2*e^2-2*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b
^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2
*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e
-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*c*d*e^3+3*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c
+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2
*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)
^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b
^2)^(1/2)*b*c*d^2*e^2-12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)
^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*
d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*
c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*d*e^3+8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*
d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1
/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(
1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*d*e^3+4*x*c^3*d^3*e-4*x
^2*b*c^2*d*e^3-12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))
/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/
2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2
*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*c*e^4+2*a*b^2*e^4+12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-
2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2
)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d
))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c^2*d^2*e^2-8*2^(1/2)*(
-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^
(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x
+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/
2)))^(1/2))*a*c^2*d^2*e^2+8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b
^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2
*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e
-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*d^2*e^2-3*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e
-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^
2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*
d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*d^2*e^2-2^(1/2)*(-
(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(
1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+
d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2
)))^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*d*e^3-2*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2
*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^
2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+
b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*c^2*d^3*e-8*2^(1/2)*(-(e*x+d
)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))
^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(
e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1
/2))*b*c^2*d^3*e+2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/
(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2
)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*
c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*b*e^4-12*x^2*a*c^2*e^4+4*x^2*b^2*c*e^4+4*x^2*c^3*d^
2*e^2-8*a^2*c*e^4+2*b^3*d*e^3+4*x*b^3*e^4-14*x*a*b*c*e^4+4*x*a*c^2*d*e^3-2*x*b^2*c*d*e^3+8*a*c^2*d^2*e^2+2*b*c
^2*d^3*e)*(c*x^2+b*x+a)^(1/2)*(e*x+d)^(1/2)/e/(4*a*c-b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x
+b*d*x+a*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c^{2} e^{2} x^{6} + 2 \,{\left (c^{2} d e + b c e^{2}\right )} x^{5} +{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{4} + a^{2} d^{2} + 2 \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{3} +{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{2} + 2 \,{\left (a b d^{2} + a^{2} d e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(c^2*e^2*x^6 + 2*(c^2*d*e + b*c*e^2)*x^5 + (c^2*d^2 + 4*b*c*d*e +
(b^2 + 2*a*c)*e^2)*x^4 + a^2*d^2 + 2*(b*c*d^2 + a*b*e^2 + (b^2 + 2*a*c)*d*e)*x^3 + (4*a*b*d*e + a^2*e^2 + (b^
2 + 2*a*c)*d^2)*x^2 + 2*(a*b*d^2 + a^2*d*e)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*(a + b*x + c*x**2)**(3/2)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

Timed out