3.684 \(\int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx\)
Optimal. Leaf size=447 \[ \frac {16 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {2 c e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{15 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}-\frac {16 c d e \sqrt {a+c x^2}}{15 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )} \]
[Out]
-2/5*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(5/2)-16/15*c*d*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)-2
/15*c*e*(-9*a*e^2+23*c*d^2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)-2/15*c^(3/2)*(-9*a*e^2+23*c*d^2)*Ell
ipticE(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)*(c*x^2/a+1)^(1/2)/(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)+16
/15*c^(3/2)*d*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)*(c*x^2/a+1)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)/(
c*x^2+a)^(1/2)
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Rubi [A] time = 0.41, antiderivative size = 447, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used =
{745, 835, 844, 719, 424, 419} \[ \frac {16 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {2 c e \sqrt {a+c x^2} \left (23 c d^2-9 a e^2\right )}{15 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}-\frac {16 c d e \sqrt {a+c x^2}}{15 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac {2 e \sqrt {a+c x^2}}{5 (d+e x)^{5/2} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^(7/2)*Sqrt[a + c*x^2]),x]
[Out]
(-2*e*Sqrt[a + c*x^2])/(5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (16*c*d*e*Sqrt[a + c*x^2])/(15*(c*d^2 + a*e^2)^2*
(d + e*x)^(3/2)) - (2*c*e*(23*c*d^2 - 9*a*e^2)*Sqrt[a + c*x^2])/(15*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) - (2*Sqrt
[-a]*c^(3/2)*(23*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)])/(15*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*
d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*c^(3/2)*d*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)])/(15*(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 745
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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*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] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])
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)^{7/2} \sqrt {a+c x^2}} \, dx &=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {(2 c) \int \frac {-\frac {5 d}{2}+\frac {3 e x}{2}}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx}{5 \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {16 c d e \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {(4 c) \int \frac {\frac {3}{4} \left (5 c d^2-3 a e^2\right )-2 c d e x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^2}\\ &=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {16 c d e \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 c e \left (23 c d^2-9 a e^2\right ) \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {(8 c) \int \frac {-\frac {1}{8} c d \left (15 c d^2-17 a e^2\right )-\frac {1}{8} c e \left (23 c d^2-9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^3}\\ &=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {16 c d e \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 c e \left (23 c d^2-9 a e^2\right ) \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (c^2 \left (23 c d^2-9 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^3}-\frac {\left (8 c^2 d\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 \left (c d^2+a e^2\right )^2}\\ &=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {16 c d e \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 c e \left (23 c d^2-9 a e^2\right ) \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (2 a c^{3/2} \left (23 c d^2-9 a e^2\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 )}{15 \sqrt {-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 (16 a c^{3/2} d \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 )}{15 \sqrt {-a} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {2 e \sqrt {a+c x^2}}{5 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {16 c d e \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 c e \left (23 c d^2-9 a e^2\right ) \sqrt {a+c x^2}}{15 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \sqrt {-a} c^{3/2} \left (23 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 )}{15 \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 {16 \sqrt {-a} c^{3/2} d \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 )}{15 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 2.45, size = 618, normalized size = 1.38 \[ \frac {2 \left (-\left (e^2 \left (a+c x^2\right ) \left (8 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (23 c d^2-9 a e^2\right )+3 \left (a e^2+c d^2\right )^2\right )\right )-\frac {c (d+e x)^2 \left (\sqrt {c} (d+e x)^{3/2} \left (-9 a^{3/2} e^3+23 \sqrt {a} c d^2 e+17 i a \sqrt {c} d e^2-15 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} (d+e x)^{3/2} \left (9 a^{3/2} e^3-23 \sqrt {a} c d^2 e-9 i a \sqrt {c} d e^2+23 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+e^2 \left (-\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}\right ) \left (-9 a^2 e^2+a c \left (23 d^2-9 e^2 x^2\right )+23 c^2 d^2 x^2\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{15 e \sqrt {a+c x^2} (d+e x)^{5/2} \left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(7/2)*Sqrt[a + c*x^2]),x]
[Out]
(2*(-(e^2*(a + c*x^2)*(3*(c*d^2 + a*e^2)^2 + 8*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(23*c*d^2 - 9*a*e^2)*(d + e*x
)^2)) - (c*(d + e*x)^2*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-9*a^2*e^2 + 23*c^2*d^2*x^2 + a*c*(23*d^2 - 9*
e^2*x^2))) + Sqrt[c]*((23*I)*c^(3/2)*d^3 - 23*Sqrt[a]*c*d^2*e - (9*I)*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*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)*Ellipt
icE[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[c]*((-15*I)*c^(3/2)*d^3 + 23*Sqrt[a]*c*d^2*e + (17*I)*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*e^3)*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]]))/(15*e*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2)*Sqrt[a + c*x^2])
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{c e^{4} x^{6} + 4 \, c d e^{3} x^{5} + 4 \, a d^{3} e x + a d^{4} + {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} + 4 \, {\left (c d^{3} e + a d e^{3}\right )} x^{3} + {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(c*x^2 + a)*sqrt(e*x + d)/(c*e^4*x^6 + 4*c*d*e^3*x^5 + 4*a*d^3*e*x + a*d^4 + (6*c*d^2*e^2 + a*e^4
)*x^4 + 4*(c*d^3*e + a*d*e^3)*x^3 + (c*d^4 + 6*a*d^2*e^2)*x^2), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(c*x^2 + a)*(e*x + d)^(7/2)), x)
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maple [B] time = 0.14, size = 3863, normalized size = 8.64 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x)
[Out]
-2/15*(-10*a*c^2*d*e^5*x^3+28*a*c^2*d^2*e^4*x^2+54*a*c^2*d^3*e^3*x+54*c^3*d^3*e^3*x^3-6*a^2*c*e^6*x^2-9*a*c^2*
e^6*x^4+23*c^3*d^2*e^4*x^4+5*a^2*c*d^2*e^4+34*a*c^2*d^4*e^2-15*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))*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)+23*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))*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)+3*a^3*e^6+34*c^3*d^4*e^2*x^2-6*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^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)-8*Ellipt
icF((-(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^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)+14*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^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)+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*a^2*c*d*e^5*(
-(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*a*c^2*d^3*e^3*(-(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*c^2*d^4*e^2*(-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)-18*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*a^2*c*d*e^5*(-(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)+28*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*a*c^2*d^3*e^3*(-(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)-8*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*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)-8*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*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*a*c*d^2*e^4*(-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)+23*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^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)+14
*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^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)+9*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^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)-6*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^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)+46*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*c^3*d^5*e*(-(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)+9*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*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)-9*Ellip
ticE((-(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*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)-8*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))*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)-9*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*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)-10*a^2*c*d*e^5*x-15*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^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)-30*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*c^3*d^5*e*(-(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))/(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^(5/2)/e
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(c*x^2 + a)*(e*x + d)^(7/2)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + c*x^2)^(1/2)*(d + e*x)^(7/2)),x)
[Out]
int(1/((a + c*x^2)^(1/2)*(d + e*x)^(7/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**(7/2)/(c*x**2+a)**(1/2),x)
[Out]
Integral(1/(sqrt(a + c*x**2)*(d + e*x)**(7/2)), x)
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