\(\int \frac {(d+e x)^{9/2}}{(a+c x^2)^{5/2}} \, dx\) [692]

Optimal result

Integrand size = 21, antiderivative size = 475 \[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x)^{3/2} \left (a e \left (c d^2+7 a e^2\right )-2 c d \left (2 c d^2+5 a e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a+c x^2}}-\frac {2 d e \left (c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {\left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} c^{5/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-1/3*(-c*d*x+a*e)*(e*x+d)^(7/2)/a/c/(c*x^2+a)^(3/2)-1/6*(e*x+d)^(3/2)*(a*e*(7*a*e^2+c*d^2)-2*c*d*(5*a*e^2+2*c*
d^2)*x)/a^2/c^2/(c*x^2+a)^(1/2)-2/3*d*e*(3*a*e^2+c*d^2)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/a^2/c^2+1/6*(-21*a^2*e^4
+15*a*c*d^2*e^2+4*c^2*d^4)*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))*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/(-a)^(3/2)/c^(5/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2
)+d*c^(1/2)))^(1/2)-2/3*d*(a*e^2+c*d^2)*(3*a*e^2+c*d^2)*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))*(1+c*x^2/a)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/
(-a)^(3/2)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {753, 833, 847, 858, 733, 435, 430} \[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} c^{5/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {(d+e x)^{3/2} \left (a e \left (7 a e^2+c d^2\right )-2 c d x \left (5 a e^2+2 c d^2\right )\right )}{6 a^2 c^2 \sqrt {a+c x^2}}-\frac {2 d e \sqrt {a+c x^2} \sqrt {d+e x} \left (3 a e^2+c d^2\right )}{3 a^2 c^2}-\frac {2 d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} c^{5/2} \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {(d+e x)^{7/2} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[In]

Int[(d + e*x)^(9/2)/(a + c*x^2)^(5/2),x]

[Out]

-1/3*((a*e - c*d*x)*(d + e*x)^(7/2))/(a*c*(a + c*x^2)^(3/2)) - ((d + e*x)^(3/2)*(a*e*(c*d^2 + 7*a*e^2) - 2*c*d
*(2*c*d^2 + 5*a*e^2)*x))/(6*a^2*c^2*Sqrt[a + c*x^2]) - (2*d*e*(c*d^2 + 3*a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])
/(3*a^2*c^2) + ((4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[S
qrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*c^(5/2)*Sqrt[(Sqrt
[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (2*d*(c*d^2 + a*e^2)*(c*d^2 + 3*a*e^2)*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]]/Sq
rt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(-a)^(3/2)*c^(5/2)*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 753

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 833

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

Rule 847

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

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 {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(d+e x)^{5/2} \left (\frac {1}{2} \left (4 c d^2+7 a e^2\right )-\frac {3}{2} c d e x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x)^{3/2} \left (a e \left (c d^2+7 a e^2\right )-2 c d \left (2 c d^2+5 a e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {\sqrt {d+e x} \left (-\frac {3}{4} a e^2 \left (c d^2-7 a e^2\right )-3 c d e \left (c d^2+3 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x)^{3/2} \left (a e \left (c d^2+7 a e^2\right )-2 c d \left (2 c d^2+5 a e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a+c x^2}}-\frac {2 d e \left (c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {2 \int \frac {\frac {3}{8} a c d e^2 \left (c d^2+33 a e^2\right )-\frac {3}{8} c e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{9 a^2 c^3} \\ & = -\frac {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x)^{3/2} \left (a e \left (c d^2+7 a e^2\right )-2 c d \left (2 c d^2+5 a e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a+c x^2}}-\frac {2 d e \left (c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {\left (d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 c^2}-\frac {\left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{12 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x)^{3/2} \left (a e \left (c d^2+7 a e^2\right )-2 c d \left (2 c d^2+5 a e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a+c x^2}}-\frac {2 d e \left (c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a^2 c^2}-\frac {\left (\left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{6 \sqrt {-a} a c^{5/2} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\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 )}{3 \sqrt {-a} a c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {(a e-c d x) (d+e x)^{7/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x)^{3/2} \left (a e \left (c d^2+7 a e^2\right )-2 c d \left (2 c d^2+5 a e^2\right ) x\right )}{6 a^2 c^2 \sqrt {a+c x^2}}-\frac {2 d e \left (c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {\left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{6 (-a)^{3/2} c^{5/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\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 )}{3 (-a)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.81 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {8 c^3 d^4 x^3-2 a^3 e^3 (19 d+7 e x)+2 a c^2 d^2 x \left (6 d^2+d e x+15 e^2 x^2\right )-2 a^2 c e \left (7 d^3-3 d^2 e x+27 d e^2 x^2+9 e^3 x^3\right )}{a^2 c^2 \left (a+c x^2\right )}+\frac {2 \left (-e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \left (a+c x^2\right )+\sqrt {c} \left (4 i c^{5/2} d^5-4 \sqrt {a} c^2 d^4 e+15 i a c^{3/2} d^3 e^2-15 a^{3/2} c d^2 e^3-21 i a^2 \sqrt {c} d e^4+21 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (4 c^2 d^4+i \sqrt {a} c^{3/2} d^3 e+15 a c d^2 e^2+33 i a^{3/2} \sqrt {c} d e^3-21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{a^2 c^3 e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{12 \sqrt {a+c x^2}} \]

[In]

Integrate[(d + e*x)^(9/2)/(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[d + e*x]*((8*c^3*d^4*x^3 - 2*a^3*e^3*(19*d + 7*e*x) + 2*a*c^2*d^2*x*(6*d^2 + d*e*x + 15*e^2*x^2) - 2*a^2
*c*e*(7*d^3 - 3*d^2*e*x + 27*d*e^2*x^2 + 9*e^3*x^3))/(a^2*c^2*(a + c*x^2)) + (2*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)
/Sqrt[c]]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*(a + c*x^2)) + Sqrt[c]*((4*I)*c^(5/2)*d^5 - 4*Sqrt[a]*c^2*
d^4*e + (15*I)*a*c^(3/2)*d^3*e^2 - 15*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*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)*Elliptic
E[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]*Sqrt[c]*e*(4*c^2*d^4 + I*Sqrt[a]*c^(3/2)*d^3*e + 15*a*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 -
21*a^2*e^4)*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*Sqrt[a]*e)/(
Sqrt[c]*d + I*Sqrt[a]*e)]))/(a^2*c^3*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(12*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(882\) vs. \(2(397)=794\).

Time = 3.54 (sec) , antiderivative size = 883, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{3 a \,c^{4}}+\frac {4 d e \left (e^{2} a -c \,d^{2}\right )}{3 c^{4}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right ) x}{12 a^{2} c^{3}}+\frac {\left (27 e^{2} a -c \,d^{2}\right ) d e}{12 c^{3} a}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {5 d \,e^{4}}{c^{2}}-\frac {2 d \left (9 a^{2} e^{4}-4 a c \,d^{2} e^{2}-c^{2} d^{4}\right )}{3 a^{2} c^{2}}+\frac {e^{2} \left (27 e^{2} a -c \,d^{2}\right ) d}{12 c^{2} a}+\frac {d \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{6 c^{2} a^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {e^{5}}{c^{2}}+\frac {\left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right ) e}{12 a^{2} c^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(883\)
default	\(\text {Expression too large to display}\)	\(3304\)

[In]

int((e*x+d)^(9/2)/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*((1/3*(a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)/a/c^4*x+4/3*d*e*
(a*e^2-c*d^2)/c^4)*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x^2+1/c*a)^2-2*(c*e*x+c*d)*(1/12*(9*a^2*e^4-15*a*c*d^2*e
^2-4*c^2*d^4)/a^2/c^3*x+1/12*(27*a*e^2-c*d^2)*d*e/c^3/a)/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)+2*(5*d*e^4/c^2-2/3*d*
(9*a^2*e^4-4*a*c*d^2*e^2-c^2*d^4)/a^2/c^2+1/12/c^2*e^2*(27*a*e^2-c*d^2)*d/a+1/6/c^2*d*(9*a^2*e^4-15*a*c*d^2*e^
2-4*c^2*d^4)/a^2)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(
1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((
x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+2*(e^5/c^2+1/12*(9*a^2
*e^4-15*a*c*d^2*e^2-4*c^2*d^4)*e/a^2/c^2)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)
^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x
+a*d)^(1/2)*((-d/e-(-a*c)^(1/2)/c)*EllipticE(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e
-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/
(-d/e-(-a*c)^(1/2)/c))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (2 \, a^{2} c^{2} d^{5} + 9 \, a^{3} c d^{3} e^{2} + 39 \, a^{4} d e^{4} + {\left (2 \, c^{4} d^{5} + 9 \, a c^{3} d^{3} e^{2} + 39 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \, {\left (2 \, a c^{3} d^{5} + 9 \, a^{2} c^{2} d^{3} e^{2} + 39 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (4 \, a^{2} c^{2} d^{4} e + 15 \, a^{3} c d^{2} e^{3} - 21 \, a^{4} e^{5} + {\left (4 \, c^{4} d^{4} e + 15 \, a c^{3} d^{2} e^{3} - 21 \, a^{2} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (4 \, a c^{3} d^{4} e + 15 \, a^{2} c^{2} d^{2} e^{3} - 21 \, a^{3} c e^{5}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} + 19 \, a^{3} c d e^{4} - {\left (4 \, c^{4} d^{4} e + 15 \, a c^{3} d^{2} e^{3} - 9 \, a^{2} c^{2} e^{5}\right )} x^{3} - {\left (a c^{3} d^{3} e^{2} - 27 \, a^{2} c^{2} d e^{4}\right )} x^{2} - {\left (6 \, a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3} - 7 \, a^{3} c e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{2} c^{5} e x^{4} + 2 \, a^{3} c^{4} e x^{2} + a^{4} c^{3} e\right )}} \]

[In]

integrate((e*x+d)^(9/2)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/18*(2*(2*a^2*c^2*d^5 + 9*a^3*c*d^3*e^2 + 39*a^4*d*e^4 + (2*c^4*d^5 + 9*a*c^3*d^3*e^2 + 39*a^2*c^2*d*e^4)*x^4
+ 2*(2*a*c^3*d^5 + 9*a^2*c^2*d^3*e^2 + 39*a^3*c*d*e^4)*x^2)*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) + 3*(4*a^2*c^2*d^4*e + 15*a^3*c*d^2*e^3 - 21
*a^4*e^5 + (4*c^4*d^4*e + 15*a*c^3*d^2*e^3 - 21*a^2*c^2*e^5)*x^4 + 2*(4*a*c^3*d^4*e + 15*a^2*c^2*d^2*e^3 - 21*
a^3*c*e^5)*x^2)*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), we
ierstrassPInverse(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*(7
*a^2*c^2*d^3*e^2 + 19*a^3*c*d*e^4 - (4*c^4*d^4*e + 15*a*c^3*d^2*e^3 - 9*a^2*c^2*e^5)*x^3 - (a*c^3*d^3*e^2 - 27
*a^2*c^2*d*e^4)*x^2 - (6*a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3 - 7*a^3*c*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(a^2
*c^5*e*x^4 + 2*a^3*c^4*e*x^2 + a^4*c^3*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(9/2)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(9/2)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(9/2)/(c*x^2 + a)^(5/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(9/2)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(9/2)/(c*x^2 + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(9/2)/(a + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^(9/2)/(a + c*x^2)^(5/2), x)