\(\int (d+e x)^{3/2} (a+c x^2)^{3/2} \, dx\) [663]

Optimal result

Integrand size = 21, antiderivative size = 497 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \sqrt {-a} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 \sqrt {c} e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

2/231*(28*c*d*e*x-3*a*e^2+c*d^2)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/c/e+2/11*e*(c*x^2+a)^(5/2)*(e*x+d)^(1/2)/c+4/11
55*(4*c^2*d^4+21*a*c*d^2*e^2-15*a^2*e^4-3*c*d*e*(-31*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/c/e^3+32/11
55*d*(-3*a*e^2+c*d^2)*(9*a*e^2+c*d^2)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a
)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^4/c^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/
2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-8/1155*(a*e^2+c*d^2)*(-15*a^2*e^4+21*a*c*d^2*e^2+4*c^2*d^4)*EllipticF(1/2*(
1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^2/a)^(1/2)
*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/c^(3/2)/e^4/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {757, 829, 858, 733, 435, 430} \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=-\frac {8 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}+\frac {32 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 \sqrt {c} e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c} \]

[In]

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

[Out]

(4*Sqrt[d + e*x]*(4*c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4 - 3*c*d*e*(c*d^2 - 31*a*e^2)*x)*Sqrt[a + c*x^2])/(11
55*c*e^3) + (2*Sqrt[d + e*x]*(c*d^2 - 3*a*e^2 + 28*c*d*e*x)*(a + c*x^2)^(3/2))/(231*c*e) + (2*e*Sqrt[d + e*x]*
(a + c*x^2)^(5/2))/(11*c) + (32*Sqrt[-a]*d*(c*d^2 - 3*a*e^2)*(c*d^2 + 9*a*e^2)*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)])/(1155*Sqrt[
c]*e^4*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)*(4*c^
2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*El
lipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(1155*c^(3/2)*e^
4*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 430

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

Rule 435

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

Rule 733

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 757

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

Rule 829

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 858

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {2 \int \frac {\left (\frac {1}{2} \left (11 c d^2-a e^2\right )+6 c d e x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 c} \\ & = \frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {8 \int \frac {\left (\frac {3}{4} a c e^2 \left (29 c d^2-3 a e^2\right )-\frac {3}{4} c^2 d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{231 c^2 e^2} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \int \frac {\frac {3}{8} a c^2 e^2 \left (c^2 d^4+114 a c d^2 e^2-15 a^2 e^4\right )-\frac {3}{2} c^3 d e \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3465 c^3 e^4} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {\left (16 d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{1155 e^4}+\frac {\left (4 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{1155 c e^4} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {\left (32 a d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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 )}{1155 \sqrt {-a} \sqrt {c} e^4 \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 (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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 )}{1155 \sqrt {-a} c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \sqrt {-a} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\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 )}{1155 \sqrt {c} e^4 \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 (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\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 )}{1155 c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.71 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.40 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (60 a^2 e^4+a c e^2 \left (47 d^2+326 d e x+195 e^2 x^2\right )+c^2 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )}{c e^3}-\frac {8 \left (4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c^2 d^4+6 a c d^2 e^2-27 a^2 e^4\right ) \left (a+c x^2\right )+4 \sqrt {c} d \left (-i c^{5/2} d^5+\sqrt {a} c^2 d^4 e-6 i a c^{3/2} d^3 e^2+6 a^{3/2} c d^2 e^3+27 i a^2 \sqrt {c} d e^4-27 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (-4 c^{5/2} d^5-i \sqrt {a} c^2 d^4 e-24 a c^{3/2} d^3 e^2-114 i a^{3/2} c d^2 e^3+108 a^2 \sqrt {c} d e^4+15 i a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{1155 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(60*a^2*e^4 + a*c*e^2*(47*d^2 + 326*d*e*x + 195*e^2*x^2) + c^2*(8*d^4 - 6*d^3*e
*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)))/(c*e^3) - (8*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c^
2*d^4 + 6*a*c*d^2*e^2 - 27*a^2*e^4)*(a + c*x^2) + 4*Sqrt[c]*d*((-I)*c^(5/2)*d^5 + Sqrt[a]*c^2*d^4*e - (6*I)*a*
c^(3/2)*d^3*e^2 + 6*a^(3/2)*c*d^2*e^3 + (27*I)*a^2*Sqrt[c]*d*e^4 - 27*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[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*ArcSinh[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]*e*(-
4*c^(5/2)*d^5 - I*Sqrt[a]*c^2*d^4*e - 24*a*c^(3/2)*d^3*e^2 - (114*I)*a^(3/2)*c*d^2*e^3 + 108*a^2*Sqrt[c]*d*e^4
+ (15*I)*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[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[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sq
rt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(1155*Sqrt[a + c*x^
2])

Maple [F(-1)]

Timed out.

hanged

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.74 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (4 \, {\left (4 \, c^{3} d^{6} + 27 \, a c^{2} d^{4} e^{2} + 234 \, a^{2} c d^{2} e^{4} - 45 \, a^{3} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 48 \, {\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} - 27 \, a^{2} c d e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (105 \, c^{3} e^{6} x^{4} + 140 \, c^{3} d e^{5} x^{3} + 8 \, c^{3} d^{4} e^{2} + 47 \, a c^{2} d^{2} e^{4} + 60 \, a^{2} c e^{6} + 5 \, {\left (c^{3} d^{2} e^{4} + 39 \, a c^{2} e^{6}\right )} x^{2} - 2 \, {\left (3 \, c^{3} d^{3} e^{3} - 163 \, a c^{2} d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{3465 \, c^{2} e^{5}} \]

[In]

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

[Out]

2/3465*(4*(4*c^3*d^6 + 27*a*c^2*d^4*e^2 + 234*a^2*c*d^2*e^4 - 45*a^3*e^6)*sqrt(c*e)*weierstrassPInverse(4/3*(c
*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 48*(c^3*d^5*e + 6*a*c^2*d^3*e
^3 - 27*a^2*c*d*e^5)*sqrt(c*e)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3
), weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)) +
3*(105*c^3*e^6*x^4 + 140*c^3*d*e^5*x^3 + 8*c^3*d^4*e^2 + 47*a*c^2*d^2*e^4 + 60*a^2*c*e^6 + 5*(c^3*d^2*e^4 + 3
9*a*c^2*e^6)*x^2 - 2*(3*c^3*d^3*e^3 - 163*a*c^2*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c^2*e^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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