3.698 \(\int \frac {1}{(d+e x)^{3/2} (a+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=532 \[ \frac {e \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{6 a^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (3 a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.53, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {741, 823, 835, 844, 719, 424, 419} \[ \frac {e \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{6 a^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (3 a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*(a + c*x^2)^(3/2)) - (a*e*(c*d^2 - 7*a*e^2) - 4*c*d*(c*d^2 +
3*a*e^2)*x)/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*
e^4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[c]*(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[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqr
t[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sq
rt[a + c*x^2]) - (2*Sqrt[c]*d*(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]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(-
a)^(3/2)*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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)])

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 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 741

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*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 823

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

Rule 835

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 4.06, size = 669, normalized size = 1.26 \[ \frac {21 a^3 e^5+c (d+e x) \left (3 a^2 e^3 (7 d-3 e x)+a c d^2 e (d+15 e x)+4 c^2 d^4 x\right )-\frac {i c (d+e x)^{3/2} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+3 a^2 c e^3 \left (7 e^2 x^2-5 d^2\right )-12 a^2 e^5 \left (a+c x^2\right )+\frac {\sqrt {a} \sqrt {c} (d+e x)^{3/2} \left (33 i a^{3/2} \sqrt {c} d e^3-21 a^2 e^4+i \sqrt {a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}-a c^2 d^2 e \left (4 d^2+15 e^2 x^2\right )+\frac {2 a c (d+e x) \left (a e^2+c d^2\right ) \left (a e (2 d-e x)+c d^2 x\right )}{a+c x^2}-4 c^3 d^4 e x^2}{6 a^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(21*a^3*e^5 - 4*c^3*d^4*e*x^2 - 12*a^2*e^5*(a + c*x^2) + 3*a^2*c*e^3*(-5*d^2 + 7*e^2*x^2) - a*c^2*d^2*e*(4*d^2
+ 15*e^2*x^2) + (2*a*c*(c*d^2 + a*e^2)*(d + e*x)*(c*d^2*x + a*e*(2*d - e*x)))/(a + c*x^2) + c*(d + e*x)*(4*c^
2*d^4*x + 3*a^2*e^3*(7*d - 3*e*x) + a*c*d^2*e*(d + 15*e*x)) - (I*c*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)*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)])/e + (Sqrt[a]*Sqrt[c]*(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)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])/(6*a^2
*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{c^{3} e^{2} x^{8} + 2 \, c^{3} d e x^{7} + 6 \, a c^{2} d e x^{5} + 6 \, a^{2} c d e x^{3} + {\left (c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{6} + 2 \, a^{3} d e x + a^{3} d^{2} + 3 \, {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{4} + {\left (3 \, a^{2} c d^{2} + a^{3} e^{2}\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + a)*sqrt(e*x + d)/(c^3*e^2*x^8 + 2*c^3*d*e*x^7 + 6*a*c^2*d*e*x^5 + 6*a^2*c*d*e*x^3 + (c^3
*d^2 + 3*a*c^2*e^2)*x^6 + 2*a^3*d*e*x + a^3*d^2 + 3*(a*c^2*d^2 + a^2*c*e^2)*x^4 + (3*a^2*c*d^2 + a^3*e^2)*x^2)
, x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
1.61Unable to transpose Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.18, size = 3322, normalized size = 6.24 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x)

[Out]

-1/6*(21*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2
))*x^2*a^3*c*e^6*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*(
(c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+3*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d
+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+
(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+21*EllipticE((-
(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^4*e^6*(-(e*x+d)/
(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-
a*c)^(1/2)*e)*e)^(1/2)-7*x^2*a*c^3*d^4*e^2-21*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c
)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a^3*c*e^6*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(
1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-21*EllipticF((-(e*x+d)/
(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^4*e^6*(-(e*x+d)/(-c*d+(-
a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/
2)*e)*e)^(1/2)+12*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)
*e))^(1/2))*x^2*a^2*c*d*e^5*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-
a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+16*EllipticF((-(e*x+d)/(-c*d+(-a*c)^
(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a*c^2*d^3*e^3*(-a*c)^(1/2)*(-(e*x+d
)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+
(-a*c)^(1/2)*e)*e)^(1/2)-4*x^3*c^4*d^5*e+21*x^4*a^2*c^2*e^6-4*x^4*c^4*d^4*e^2+35*x^2*a^3*c*e^6-15*x^4*a*c^3*d^
2*e^4-12*x^3*a^2*c^2*d*e^5-16*x^3*a*c^3*d^3*e^3-14*x*a^3*c*d*e^5-36*x^2*a^2*c^2*d^2*e^4-20*x*a^2*c^2*d^3*e^3-2
5*a^3*c*d^2*e^4-5*a^2*c^2*d^4*e^2+6*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)
/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a^2*c^2*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2)
)/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-19*EllipticE((-(e*x+d)/(-c*
d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a*c^3*d^4*e^2*(-(e*x+d)/(-
c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*
c)^(1/2)*e)*e)^(1/2)-18*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)
^(1/2)*e))^(1/2))*x^2*a^2*c^2*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c
)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+3*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2
)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a*c^3*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1
/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e
)^(1/2)+4*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/
2))*x^2*c^3*d^5*e*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)
*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+16*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c
)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*c*d^3*e^3*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)
^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e
)*e)^(1/2)+4*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^
(1/2))*a*c^2*d^5*e*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2
)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+12*a^4*e^6+6*EllipticE((-(e*x+d)/(-c*d+(-a*c)
^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^3*c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(
1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*
e)^(1/2)-19*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(
1/2))*a^2*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1
/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-18*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(
-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^3*c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-
c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+12*Elliptic
F((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^3*d*e^5*(-a*
c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a
*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-6*x*a*c^3*d^5*e-4*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)
,(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c^3*d^6*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*
x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-4*EllipticE((
-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*c^4*d^6*(-(e*
x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c
*d+(-a*c)^(1/2)*e)*e)^(1/2))/(e*x+d)^(1/2)/(a*e^2+c*d^2)^3/a^2/(c*x^2+a)^(3/2)/e

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^(3/2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^(5/2)*(d + e*x)^(3/2)),x)

[Out]

int(1/((a + c*x^2)^(5/2)*(d + e*x)^(3/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**(3/2)), x)

________________________________________________________________________________________