\(\int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx\) [662]

Optimal result

Integrand size = 21, antiderivative size = 444 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \sqrt {-a} c^{3/2} \left (c d^2-3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]

[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)*(1+c*x^2/a)^(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)*(1+c*x^2
/a)^(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)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 849, 858, 733, 435, 430} \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {4 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 c \sqrt {a+c x^2} \left (c d^2-3 a e^2\right )}{15 e \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {4 c d \sqrt {a+c x^2}}{15 e (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}} \]

[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 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*} \text {integral}& = -\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 ) \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 )}{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 ) \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 )}{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*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.24 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-e^2 \left (a+c x^2\right ) \left (3 a^2 e^4-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )+2 a c e^2 \left (5 d^2+5 d e x+3 e^2 x^2\right )\right )+\frac {2 \left (-c e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2-3 a e^2\right ) (d+e x)^2 \left (a+c x^2\right )+c^{3/2} \left (i c^{3/2} d^3-\sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} c^{3/2} e \left (c d^2+4 i \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{15 e^3 \left (c d^2+a e^2\right )^2 (d+e x)^{5/2} \sqrt {a+c x^2}} \]

[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*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 - 3*a*e^2)*(d + e*x)^2*(a + c*x^2)) + c^(3/2)*(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)^(7/2)*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]*c^(3/2)*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)^(7/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (S
qrt[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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(792\) vs. \(2(366)=732\).

Time = 2.19 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.79

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 e^{3} \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 \left (c e \,x^{2}+a e \right ) c \left (3 e^{2} a -c \,d^{2}\right )}{15 e^{2} \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {2 d \,c^{2}}{15 e^{2} \left (e^{2} a +c \,d^{2}\right )}+\frac {2 d \,c^{2} \left (3 e^{2} a -c \,d^{2}\right )}{15 e^{2} \left (e^{2} a +c \,d^{2}\right )^{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 {4 c^{2} \left (3 e^{2} a -c \,d^{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 )}{15 e \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(793\)
default	\(\text {Expression too large to display}\)	\(3411\)

[In]

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

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/5/e^4*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3+
4/15*c/e^3/(a*e^2+c*d^2)*d*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2-4/15*(c*e*x^2+a*e)/e^2/(a*e^2+c*d^2)^2*
c*(3*a*e^2-c*d^2)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(2/15*d/e^2*c^2/(a*e^2+c*d^2)+2/15*d/e^2*c^2*(3*a*e^2-c*d^2)
/(a*e^2+c*d^2)^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))+4/15/e*c^2*(3*a*e^2-c*
d^2)/(a*e^2+c*d^2)^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 = 569, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (2 \, {\left (c^{2} d^{6} + 9 \, a c d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + 9 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (c^{2} d^{4} e^{2} + 9 \, a c d^{2} e^{4}\right )} x^{2} + 3 \, {\left (c^{2} d^{5} e + 9 \, a c d^{3} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (c^{2} d^{5} e - 3 \, a c d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} - 3 \, a c e^{6}\right )} x^{3} + 3 \, {\left (c^{2} d^{3} e^{3} - 3 \, a c d e^{5}\right )} x^{2} + 3 \, {\left (c^{2} d^{4} e^{2} - 3 \, a c d^{2} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (c^{2} d^{4} e^{2} - 10 \, a c d^{2} e^{4} - 3 \, a^{2} e^{6} + 2 \, {\left (c^{2} d^{2} e^{4} - 3 \, a c e^{6}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{3} - 5 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, {\left (c^{2} d^{7} e^{3} + 2 \, a c d^{5} e^{5} + a^{2} d^{3} e^{7} + {\left (c^{2} d^{4} e^{6} + 2 \, a c d^{2} e^{8} + a^{2} e^{10}\right )} x^{3} + 3 \, {\left (c^{2} d^{5} e^{5} + 2 \, a c d^{3} e^{7} + a^{2} d e^{9}\right )} x^{2} + 3 \, {\left (c^{2} d^{6} e^{4} + 2 \, a c d^{4} e^{6} + a^{2} d^{2} e^{8}\right )} x\right )}} \]

[In]

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

[Out]

2/45*(2*(c^2*d^6 + 9*a*c*d^4*e^2 + (c^2*d^3*e^3 + 9*a*c*d*e^5)*x^3 + 3*(c^2*d^4*e^2 + 9*a*c*d^2*e^4)*x^2 + 3*(
c^2*d^5*e + 9*a*c*d^3*e^3)*x)*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) + 6*(c^2*d^5*e - 3*a*c*d^3*e^3 + (c^2*d^2*e^4 - 3*a*c*e^6)*x^3 + 3*(c^2*d^3
*e^3 - 3*a*c*d*e^5)*x^2 + 3*(c^2*d^4*e^2 - 3*a*c*d^2*e^4)*x)*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), 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*(c^2*d^4*e^2 - 10*a*c*d^2*e^4 - 3*a^2*e^6 + 2*(c^2*d^2*e^4 - 3*a*c*
e^6)*x^2 + 2*(3*c^2*d^3*e^3 - 5*a*c*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c^2*d^7*e^3 + 2*a*c*d^5*e^5 + a^
2*d^3*e^7 + (c^2*d^4*e^6 + 2*a*c*d^2*e^8 + a^2*e^10)*x^3 + 3*(c^2*d^5*e^5 + 2*a*c*d^3*e^7 + a^2*d*e^9)*x^2 + 3
*(c^2*d^6*e^4 + 2*a*c*d^4*e^6 + a^2*d^2*e^8)*x)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)