\(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [667]

Optimal result

Integrand size = 21, antiderivative size = 358 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {4 c (4 d+e x) \sqrt {a+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {32 \sqrt {-a} c^{3/2} d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (4 c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-2/3*(c*x^2+a)^(3/2)/e/(e*x+d)^(3/2)+4/3*c*(e*x+4*d)*(c*x^2+a)^(1/2)/e^3/(e*x+d)^(1/2)+32/3*c^(3/2)*d*Elliptic
E(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^4/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-8/3*(a*e^2+4*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))*(-a)^(1/2)*c
^(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^4/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 827, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {32 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+4 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^4 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {4 c \sqrt {a+c x^2} (4 d+e x)}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]

[In]

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

[Out]

(4*c*(4*d + e*x)*Sqrt[a + c*x^2])/(3*e^3*Sqrt[d + e*x]) - (2*(a + c*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + (32*Sq
rt[-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^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]
) - (8*Sqrt[-a]*Sqrt[c]*(4*c*d^2 + 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*e^4*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 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[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 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {(2 c) \int \frac {x \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx}{e} \\ & = \frac {4 c (4 d+e x) \sqrt {a+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {(4 c) \int \frac {-a e+4 c d x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^3} \\ & = \frac {4 c (4 d+e x) \sqrt {a+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {\left (16 c^2 d\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{3 e^4}+\frac {\left (4 c \left (4 c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^4} \\ & = \frac {4 c (4 d+e x) \sqrt {a+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {\left (32 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^4 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (8 a \sqrt {c} \left (4 c d^2+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 ) \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^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {4 c (4 d+e x) \sqrt {a+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {32 \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^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (4 c d^2+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 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.60 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {a+c x^2} \left (-a e^2+c \left (8 d^2+10 d e x+e^2 x^2\right )\right )}{3 e^3 (d+e x)^{3/2}}-\frac {8 c \left (4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )+4 \sqrt {c} d \left (-i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} e \left (4 \sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{3 e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {d+e x} \sqrt {a+c x^2}} \]

[In]

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

[Out]

(2*Sqrt[a + c*x^2]*(-(a*e^2) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(3*e^3*(d + e*x)^(3/2)) - (8*c*(4*d*e^2*Sqrt[-
d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + 4*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*(4*
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^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(726\) vs. \(2(286)=572\).

Time = 4.37 (sec) , antiderivative size = 727, normalized size of antiderivative = 2.03

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {16 \left (c e \,x^{2}+a e \right ) c d}{3 e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{3}}+\frac {2 \left (\frac {c \left (2 e^{2} a +3 c \,d^{2}\right )}{e^{4}}-\frac {\left (e^{2} a +c \,d^{2}\right ) c}{3 e^{4}}-\frac {8 c^{2} d^{2}}{3 e^{4}}-\frac {a c}{3 e^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}-\frac {32 c^{2} d \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{3 e^{3} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(727\)
default	\(\text {Expression too large to display}\)	\(1597\)
risch	\(\text {Expression too large to display}\)	\(2126\)

[In]

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

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/3*(a*e^2+c*d^2)/e^5*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1
/2)/(x+d/e)^2+16/3*(c*e*x^2+a*e)/e^4*c*d/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/3*c/e^3*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(
1/2)+2*(c*(2*a*e^2+3*c*d^2)/e^4-1/3*(a*e^2+c*d^2)*c/e^4-8/3*c^2/e^4*d^2-1/3/e^2*a*c)*(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))-32/3*c^2/e^3*d*(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 = 0.09 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (4 \, {\left (4 \, c d^{4} + 3 \, a d^{2} e^{2} + {\left (4 \, c d^{2} e^{2} + 3 \, a e^{4}\right )} x^{2} + 2 \, {\left (4 \, c d^{3} e + 3 \, a d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 48 \, {\left (c d e^{3} x^{2} + 2 \, c d^{2} e^{2} x + c d^{3} e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (c e^{4} x^{2} + 10 \, c d e^{3} x + 8 \, c d^{2} e^{2} - a e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]

[In]

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

[Out]

2/9*(4*(4*c*d^4 + 3*a*d^2*e^2 + (4*c*d^2*e^2 + 3*a*e^4)*x^2 + 2*(4*c*d^3*e + 3*a*d*e^3)*x)*sqrt(c*e)*weierstra
ssPInverse(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) + 48*(c*d*e^3*
x^2 + 2*c*d^2*e^2*x + c*d^3*e)*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*e^4*x^2 + 10*c*d*e^3*x + 8*c*d^2*e^2 - a*e^4)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(e^7*x^2 + 2*d*e^
6*x + d^2*e^5)

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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