3.7.61 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx\) [661]

Optimal. Leaf size=366 \[ -\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {4 \sqrt {-a} c^{3/2} d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^2 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-2/3*(c*x^2+a)^(1/2)/e/(e*x+d)^(3/2)+4/3*c*d*(c*x^2+a)^(1/2)/e/(a*e^2+c*d^2)/(e*x+d)^(1/2)+4/3*c^(3/2)*d*Ellip
ticE(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)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-4/
3*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^(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/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 849, 858, 733, 435, 430} \begin {gather*} \frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\text {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 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\text {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 e^2 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + c*x^2])/(3*e*(d + e*x)^(3/2)) + (4*c*d*Sqrt[a + c*x^2])/(3*e*(c*d^2 + a*e^2)*Sqrt[d + e*x]) + (4*
Sqrt[-a]*c^(3/2)*d*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)])/(3*e^2*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e
)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*Sqrt[c]*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*e^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 747

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 849

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

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Mathematica [C] Result contains complex when optimal does not.
time = 20.96, size = 504, normalized size = 1.38 \begin {gather*} \frac {2 \sqrt {a+c x^2} \left (-a e^2+c d (d+2 e x)\right )}{3 \left (c d^2 e+a e^3\right ) (d+e x)^{3/2}}-\frac {4 c \left (d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )+\sqrt {c} d \left (-i \sqrt {c} d+\sqrt {a} e\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 \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 {a} e \left (\sqrt {c} d+i \sqrt {a} e\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} 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 )}{3 e^3 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1308\) vs. \(2(294)=588\).
time = 0.46, size = 1309, normalized size = 3.58 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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(2*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2
)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c
)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*e^4*x+2*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-
(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*d^2*e^2*x-2*(-(e*x
+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-
a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2
)*e+c*d))^(1/2))*a*c*d*e^3*x-2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+
c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2
),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^2*d^3*e*x+2*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c
*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2
)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*d*e^
3+2*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1
/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*d^3*e-2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))
*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)
^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c*d^2*e^2-2*(-(e*x+d)*c/((-a*c)^(1/
2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d)
)^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*
c^2*d^4-2*c^2*d*x^3*e^3+a*c*e^4*x^2-d^2*e^2*x^2*c^2-2*a*c*d*e^3*x+a^2*e^4-a*c*d^2*e^2)/(c*x^2+a)^(1/2)/(a*e^2+
c*d^2)/e^3/(e*x+d)^(3/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.36, size = 315, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (2 \, {\left (2 \, c d^{3} x e + c d^{4} + 3 \, a x^{2} e^{4} + 6 \, a d x e^{3} + {\left (c d^{2} x^{2} + 3 \, a d^{2}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 6 \, {\left (c d x^{2} e^{3} + 2 \, c d^{2} x e^{2} + c d^{3} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (2 \, c d x e^{3} + c d^{2} e^{2} - a e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{9 \, {\left (2 \, c d^{3} x e^{4} + c d^{4} e^{3} + a x^{2} e^{7} + 2 \, a d x e^{6} + {\left (c d^{2} x^{2} + a d^{2}\right )} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(2*(2*c*d^3*x*e + c*d^4 + 3*a*x^2*e^4 + 6*a*d*x*e^3 + (c*d^2*x^2 + 3*a*d^2)*e^2)*sqrt(c)*e^(1/2)*weierstra
ssPInverse(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3)/c, 1/3*(3*x*e + d)*e^(-1)) + 6*(c*
d*x^2*e^3 + 2*c*d^2*x*e^2 + c*d^3*e)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*
d^3 + 9*a*d*e^2)*e^(-3)/c, weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3
)/c, 1/3*(3*x*e + d)*e^(-1))) + 3*(2*c*d*x*e^3 + c*d^2*e^2 - a*e^4)*sqrt(c*x^2 + a)*sqrt(x*e + d))/(2*c*d^3*x*
e^4 + c*d^4*e^3 + a*x^2*e^7 + 2*a*d*x*e^6 + (c*d^2*x^2 + a*d^2)*e^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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