3.1476 \(\int \frac{(A+B x) (a+c x^2)^{3/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=498 \[ \frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^5 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (36 A c d e \left (2 a e^2+c d^2\right )-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{315 e^4}-\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2} \]

[Out]

(-4*Sqrt[d + e*x]*(32*B*c*d^3 - 36*A*c*d^2*e + 33*a*B*d*e^2 - 45*a*A*e^3 - 3*e*(8*B*c*d^2 - 9*A*c*d*e + 7*a*B*
e^2)*x)*Sqrt[a + c*x^2])/(315*e^4) - (2*Sqrt[d + e*x]*(8*B*d - 9*A*e - 7*B*e*x)*(a + c*x^2)^(3/2))/(63*e^2) +
(8*Sqrt[-a]*(36*A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4))*Sqrt[d + e*x]*Sqrt[1
+ (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/
(315*Sqrt[c]*e^5*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (8*Sqrt[-a]*(c*d^2 + a*
e^2)*(32*B*c*d^3 - 36*A*c*d^2*e + 33*a*B*d*e^2 - 45*a*A*e^3)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)
]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d -
a*e)])/(315*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.553677, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {815, 844, 719, 424, 419} \[ \frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (36 A c d e \left (2 a e^2+c d^2\right )-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-3 e x \left (7 a B e^2-9 A c d e+8 B c d^2\right )-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{315 e^4}+\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (-45 a A e^3+33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^5 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[d + e*x]*(32*B*c*d^3 - 36*A*c*d^2*e + 33*a*B*d*e^2 - 45*a*A*e^3 - 3*e*(8*B*c*d^2 - 9*A*c*d*e + 7*a*B*
e^2)*x)*Sqrt[a + c*x^2])/(315*e^4) - (2*Sqrt[d + e*x]*(8*B*d - 9*A*e - 7*B*e*x)*(a + c*x^2)^(3/2))/(63*e^2) +
(8*Sqrt[-a]*(36*A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4))*Sqrt[d + e*x]*Sqrt[1
+ (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/
(315*Sqrt[c]*e^5*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (8*Sqrt[-a]*(c*d^2 + a*
e^2)*(32*B*c*d^3 - 36*A*c*d^2*e + 33*a*B*d*e^2 - 45*a*A*e^3)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)
]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d -
a*e)])/(315*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 815

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

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 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{(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx &=-\frac{2 \sqrt{d+e x} (8 B d-9 A e-7 B e x) \left (a+c x^2\right )^{3/2}}{63 e^2}+\frac{4 \int \frac{\left (-\frac{1}{2} a c e (B d-9 A e)+\frac{1}{2} c \left (8 B c d^2-9 A c d e+7 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e^2}\\ &=-\frac{4 \sqrt{d+e x} \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3-3 e \left (8 B c d^2-9 A c d e+7 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^4}-\frac{2 \sqrt{d+e x} (8 B d-9 A e-7 B e x) \left (a+c x^2\right )^{3/2}}{63 e^2}+\frac{16 \int \frac{-\frac{1}{4} a c^2 e \left (8 B c d^3-9 A c d^2 e+12 a B d e^2-45 a A e^3\right )-\frac{1}{4} c^2 \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{315 c^2 e^4}\\ &=-\frac{4 \sqrt{d+e x} \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3-3 e \left (8 B c d^2-9 A c d e+7 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^4}-\frac{2 \sqrt{d+e x} (8 B d-9 A e-7 B e x) \left (a+c x^2\right )^{3/2}}{63 e^2}-\frac{\left (4 \left (c d^2+a e^2\right ) \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{315 e^5}-\frac{\left (4 \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{315 e^5}\\ &=-\frac{4 \sqrt{d+e x} \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3-3 e \left (8 B c d^2-9 A c d e+7 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^4}-\frac{2 \sqrt{d+e x} (8 B d-9 A e-7 B e x) \left (a+c x^2\right )^{3/2}}{63 e^2}-\frac{\left (8 a \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{315 \sqrt{-a} \sqrt{c} e^5 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}-\frac{\left (8 a \left (c d^2+a e^2\right ) \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{315 \sqrt{-a} \sqrt{c} e^5 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=-\frac{4 \sqrt{d+e x} \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3-3 e \left (8 B c d^2-9 A c d e+7 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^4}-\frac{2 \sqrt{d+e x} (8 B d-9 A e-7 B e x) \left (a+c x^2\right )^{3/2}}{63 e^2}+\frac{8 \sqrt{-a} \left (36 A c d e \left (c d^2+2 a e^2\right )-B \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^5 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{8 \sqrt{-a} \left (c d^2+a e^2\right ) \left (32 B c d^3-36 A c d^2 e+33 a B d e^2-45 a A e^3\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^5 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 5.5342, size = 730, normalized size = 1.47 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (135 a A e^3+a B e^2 (77 e x-106 d)+9 A c e \left (8 d^2-6 d e x+5 e^2 x^2\right )+B c \left (48 d^2 e x-64 d^3-40 d e^2 x^2+35 e^3 x^3\right )\right )}{e^4}-\frac{8 \left (-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (B \left (21 i a^{3/2} e^3+24 i \sqrt{a} c d^2 e-33 a \sqrt{c} d e^2-32 c^{3/2} d^3\right )+9 A \sqrt{c} e \left (-3 i \sqrt{a} \sqrt{c} d e+5 a e^2+4 c d^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (36 A c d e \left (2 a e^2+c d^2\right )-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )\right )+\sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (36 A c d e \left (2 a e^2+c d^2\right )-B \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e^6 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{315 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(135*a*A*e^3 + a*B*e^2*(-106*d + 77*e*x) + 9*A*c*e*(8*d^2 - 6*d*e*x + 5*e^2*x^2
) + B*c*(-64*d^3 + 48*d^2*e*x - 40*d*e^2*x^2 + 35*e^3*x^3)))/e^4 - (8*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3
6*A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4))*(a + c*x^2) + Sqrt[c]*((-I)*Sqrt[c
]*d + Sqrt[a]*e)*(36*A*c*d*e*(c*d^2 + 2*a*e^2) - B*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4))*Sqrt[(e*((I*Sqr
t[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*Ar
cSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] -
Sqrt[a]*Sqrt[c]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(9*A*Sqrt[c]*e*(4*c*d^2 - (3*I)*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2) + B
*(-32*c^(3/2)*d^3 + (24*I)*Sqrt[a]*c*d^2*e - 33*a*Sqrt[c]*d*e^2 + (21*I)*a^(3/2)*e^3))*Sqrt[(e*((I*Sqrt[a])/Sq
rt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sq
rt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*e^6*S
qrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(315*Sqrt[a + c*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.04, size = 3113, normalized size = 6.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)/c*(5*B*x^5*c^3*d*e^5+180*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x
+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*
x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a^2*e^6-18
*A*x*a*c^2*d^2*e^4+29*B*x*a^2*c*d*e^5+16*B*x*a*c^2*d^3*e^3+34*B*x^3*a*c^2*d*e^5-126*A*x^2*a*c^2*d*e^5+98*B*x^2
*a*c^2*d^2*e^4-35*B*x^6*c^3*e^6-45*A*x^5*c^3*e^6-18*A*x^3*c^3*d^2*e^4+16*B*x^3*c^3*d^3*e^3-72*A*x^2*c^3*d^3*e^
3-77*B*x^2*a^2*c*e^6+64*B*x^2*c^3*d^4*e^2-135*A*x*a^2*c*e^6+9*A*x^4*c^3*d*e^5-112*B*x^4*a*c^2*e^6-8*B*x^4*c^3*
d^2*e^4-180*A*x^3*a*c^2*e^6-135*A*a^2*c*d*e^5-72*A*a*c^2*d^3*e^3+106*B*a^2*c*d^2*e^4+64*B*a*c^2*d^4*e^2+356*B*
(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))
*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c
)^(1/2)*e+c*d))^(1/2))*a*c^2*d^4*e^2+144*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*
e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*c^2*d^4*e^2+108*A*(-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1
/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d
))^(1/2))*a^2*c*d*e^5+108*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d
))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(
-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^3*e^3-288*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)
*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*Elliptic
E((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d*e^5-432*
A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2
))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a
*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^3*e^3-132*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((
-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2
)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a^2*d*e^5-128*B*(-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1
/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d
))^(1/2))*(-a*c)^(1/2)*c^2*d^5*e-180*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^
(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*
d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^2*e^4-96*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c
*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2
)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*
d^4*e^2+312*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x
+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)
*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^2*e^4-84*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^
(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*e^6+84*B*(-(e*x+d)*c/((-a*c
)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e
-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1
/2))*a^3*e^6+128*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*
((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^
(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^3*d^6-144*A*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^
(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^3*d^5*e+324*A*(-(e*x+d)*c/((-
a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2
)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))
^(1/2))*(-a*c)^(1/2)*a*c*d^2*e^4-260*B*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^
(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*
d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a*c*d^3*e^3)/e^6/(c*e*x^3+c*d*x^2+a
*e*x+a*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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