### 3.2441 $$\int (d+e x)^{3/2} \sqrt{a+b x+c x^2} \, dx$$

Optimal. Leaf size=581 $\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}$

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 - 4*b^2*e^2 + c*e*(9*b*d - 5*a*e) + 12*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])
/(105*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/(7*c) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(3
*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipt
icE[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)])/(105*c^3*e^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*
x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*a*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))]*Ellipt
icF[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)])/(105*c^3*e^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.868146, antiderivative size = 581, 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 = {742, 814, 843, 718, 424, 419} $\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 - 4*b^2*e^2 + c*e*(9*b*d - 5*a*e) + 12*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])
/(105*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/(7*c) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(3
*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipt
icE[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)])/(105*c^3*e^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*
x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*a*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))]*Ellipt
icF[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)])/(105*c^3*e^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, 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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 814

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 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 (d+e x)^{3/2} \sqrt{a+b x+c x^2} \, dx &=\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (\frac{1}{2} \left (7 c d^2-e (3 b d+a e)\right )+2 e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{7 c}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{4 \int \frac{-\frac{1}{4} e \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )-5 c (b d-2 a e) \left (7 c d^2-e (3 b d+a e)\right )\right )+\frac{1}{4} e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^2 e^2}+-\frac{\left (4 \left (-\frac{1}{4} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac{1}{4} e^2 \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )-5 c (b d-2 a e) \left (7 c d^2-e (3 b d+a e)\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 e^3}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 )}{105 c^3 e^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 (8 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{1}{4} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac{1}{4} e^2 \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )-5 c (b d-2 a e) \left (7 c d^2-e (3 b d+a e)\right )\right )\right ) \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 )}{105 c^3 e^3 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2-4 b^2 e^2+c e (9 b d-5 a e)+12 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 )}{105 c^3 e^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{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 a c e^2\right ) \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 )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 13.0666, size = 5328, normalized size = 9.17 $\text{Result too large to show}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.398, size = 6516, normalized size = 11.2 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out