3.662 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=444 \[ -\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {4 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 c \sqrt {a+c x^2} \left (c d^2-3 a e^2\right )}{15 e \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {4 c d \sqrt {a+c x^2}}{15 e (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}} \]

[Out]

-2/5*(c*x^2+a)^(1/2)/e/(e*x+d)^(5/2)+4/15*c*d*(c*x^2+a)^(1/2)/e/(a*e^2+c*d^2)/(e*x+d)^(3/2)+4/15*c*(-3*a*e^2+c
*d^2)*(c*x^2+a)^(1/2)/e/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+4/15*c^(3/2)*(-3*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)*(c*x^2/a+1)^(1
/2)/e^2/(a*e^2+c*d^2)^2/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-4/15*c^(3/2)*d*Ellipt
icF(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)/e^2/(a*e^2+c*d^2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.38, antiderivative size = 444, 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 = {733, 835, 844, 719, 424, 419} \[ -\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {4 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 c \sqrt {a+c x^2} \left (c d^2-3 a e^2\right )}{15 e \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {4 c d \sqrt {a+c x^2}}{15 e (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^2]/(d + e*x)^(7/2),x]

[Out]

(-2*Sqrt[a + c*x^2])/(5*e*(d + e*x)^(5/2)) + (4*c*d*Sqrt[a + c*x^2])/(15*e*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) +
(4*c*(c*d^2 - 3*a*e^2)*Sqrt[a + c*x^2])/(15*e*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) + (4*Sqrt[-a]*c^(3/2)*(c*d^2 -
3*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*e^2*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt
[a + c*x^2]) - (4*Sqrt[-a]*c^(3/2)*d*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)])/(15*e^2*(c*d^2 +
a*e^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 733

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

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

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Mathematica [C]  time = 2.11, size = 602, normalized size = 1.36 \[ \frac {2 \left (-e^2 \left (a+c x^2\right ) \left (3 a^2 e^4+2 a c e^2 \left (5 d^2+5 d e x+3 e^2 x^2\right )-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )\right )+\frac {2 c (d+e x)^2 \left (\sqrt {c} (d+e x)^{3/2} \left (3 a^{3/2} e^3-\sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2+i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+e^2 \left (-\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}\right ) \left (-3 a^2 e^2+a c \left (d^2-3 e^2 x^2\right )+c^2 d^2 x^2\right )+\sqrt {a} \sqrt {c} e (d+e x)^{3/2} \left (4 i \sqrt {a} \sqrt {c} d e-3 a e^2+c d^2\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{15 e^3 \sqrt {a+c x^2} (d+e x)^{5/2} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^2]/(d + e*x)^(7/2),x]

[Out]

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

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fricas [F]  time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + a)/(e*x + d)^(7/2), x)

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maple [B]  time = 0.13, size = 3409, normalized size = 7.68 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(1/2)/(e*x+d)^(7/2),x)

[Out]

2/15*(-10*a*c^2*d*e^5*x^3-8*a*c^2*d^2*e^4*x^2+6*a*c^2*d^3*e^3*x+6*c^3*d^3*e^3*x^3-9*a^2*c*e^6*x^2-6*a*c^2*e^6*
x^4+2*c^3*d^2*e^4*x^4-10*a^2*c*d^2*e^4+a*c^2*d^4*e^2+2*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*e^6*a^3+x^2*c^3*d^
4*e^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)-2*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^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*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)+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^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)-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*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)-12*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)-8*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)-2*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)-2*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)-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*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)+2*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)-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^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)+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^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)+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*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)+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^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)-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*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)-2*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)-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^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*x*a^2*c*d*e^5)/(c*x^2+a)^(1/2)/(
a*e^2+c*d^2)^2/e^3/(e*x+d)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate(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 \[ \int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(1/2)/(d + e*x)^(7/2),x)

[Out]

int((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 \[ \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**(7/2), x)

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