\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\) [672]

Optimal result

Integrand size = 21, antiderivative size = 457 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {16 \sqrt {-a} \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-2*(c*x^2+a)^(5/2)/e/(e*x+d)^(1/2)-20/63*c*(-7*e*x+8*d)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e^3-8/63*c*(d*(33*a*e^2+
32*c*d^2)-3*e*(7*a*e^2+8*c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5-16/63*(21*a^2*e^4+57*a*c*d^2*e^2+32*c^2*d
^4)*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
)*c^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^6/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)
+16/63*d*(a*e^2+c*d^2)*(33*a*e^2+32*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^6/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 457, 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, 829, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {16 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {16 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (33 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {8 c \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{63 e^5}-\frac {20 c \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]

[In]

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

[Out]

(-8*c*Sqrt[d + e*x]*(d*(32*c*d^2 + 33*a*e^2) - 3*e*(8*c*d^2 + 7*a*e^2)*x)*Sqrt[a + c*x^2])/(63*e^5) - (20*c*(8
*d - 7*e*x)*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(63*e^3) - (2*(a + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) - (16*Sqrt[-a]
*Sqrt[c]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*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)])/(63*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt
[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*Sqrt[c]*d*(c*d^2 + a*e^2)*(32*c*d^2 + 33*a*e^2)*Sqrt[(Sqr
t[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)])/(63*e^6*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 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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 )^{5/2}}{e \sqrt {d+e x}}+\frac {(10 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e} \\ & = -\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {40 \int \frac {\left (-\frac {1}{2} a c d e+\frac {1}{2} c \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{21 e^3} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {32 \int \frac {-a c^2 d e \left (2 c d^2+3 a e^2\right )+\frac {1}{4} c^2 \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{63 c e^5} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (8 c d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{63 e^6}+\frac {\left (8 c \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{63 e^6} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (16 a \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\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 )}{63 \sqrt {-a} e^6 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (16 a \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 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 )}{63 \sqrt {-a} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {16 \sqrt {-a} \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\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 )}{63 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 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 )}{63 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.27 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (63 a^2 e^4+2 a c e^2 \left (106 d^2+29 d e x-14 e^2 x^2\right )+c^2 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )}{e^5 (d+e x)}+\frac {16 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{5/2} d^5+32 \sqrt {a} c^2 d^4 e-57 i a c^{3/2} d^3 e^2+57 a^{3/2} c d^2 e^3-21 i a^2 \sqrt {c} d e^4+21 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c^2 d^4+8 i \sqrt {a} c^{3/2} d^3 e+57 a c d^2 e^2+12 i a^{3/2} \sqrt {c} d e^3+21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^7 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{63 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*((-2*(a + c*x^2)*(63*a^2*e^4 + 2*a*c*e^2*(106*d^2 + 29*d*e*x - 14*e^2*x^2) + c^2*(128*d^4 + 32*
d^3*e*x - 16*d^2*e^2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)))/(e^5*(d + e*x)) + (16*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt
[c]]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*(a + c*x^2) + Sqrt[c]*((-32*I)*c^(5/2)*d^5 + 32*Sqrt[a]*c^2*d^
4*e - (57*I)*a*c^(3/2)*d^3*e^2 + 57*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*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*(32*c^2*d^4 + (8*I)*Sqrt[a]*c^(3/2)*d^3*e + 57*a*c*d^2*e^2 + (12*I)*a^(3/2)*Sqrt[c]*d*e^
3 + 21*a^2*e^4)*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)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(63*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1262\) vs. \(2(381)=762\).

Time = 4.90 (sec) , antiderivative size = 1263, normalized size of antiderivative = 2.76

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1263\)
default	\(\text {Expression too large to display}\)	\(1736\)
risch	\(\text {Expression too large to display}\)	\(1745\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^(3/2),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*(c*e*x^2+a*e)*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^6/
((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/9*c^2/e^2*x^3*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)-34/63*c^2/e^3*d*x^2*(c*e*x^3+c
*d*x^2+a*e*x+a*d)^(1/2)+2/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3/e^3*d^2)/c/e*x*(c*e*x^3+c*d*x^2+a*e
*x+a*d)^(1/2)+2/3*(-c^2*d/e^4*(3*a*e^2+c*d^2)+43/63*d/e^2*c^2*a-4/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21
*c^3/e^3*d^2)/e*d)/c/e*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(-c*d*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4)/e^6+(a^2*e^
4+2*a*c*d^2*e^2+c^2*d^4)*c/e^6*d-2/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3/e^3*d^2)/c/e*a*d-1/3*(-c^2
*d/e^4*(3*a*e^2+c*d^2)+43/63*d/e^2*c^2*a-4/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3/e^3*d^2)/e*d)/c*a)
*(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))+2*(c/e^5*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d
^4)+(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)*c/e^5+68/63*d^2/e^3*c^2*a-3/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c
^3/e^3*d^2)/c*a-2/3*(-c^2*d/e^4*(3*a*e^2+c*d^2)+43/63*d/e^2*c^2*a-4/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/
21*c^3/e^3*d^2)/e*d)/e*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 = 0.11 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (32 \, c^{2} d^{6} + 81 \, a c d^{4} e^{2} + 57 \, a^{2} d^{2} e^{4} + {\left (32 \, c^{2} d^{5} e + 81 \, a c d^{3} e^{3} + 57 \, a^{2} d e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{2} d^{5} e + 57 \, a c d^{3} e^{3} + 21 \, a^{2} d e^{5} + {\left (32 \, c^{2} d^{4} e^{2} + 57 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (7 \, c^{2} e^{6} x^{4} - 10 \, c^{2} d e^{5} x^{3} - 128 \, c^{2} d^{4} e^{2} - 212 \, a c d^{2} e^{4} - 63 \, a^{2} e^{6} + 4 \, {\left (4 \, c^{2} d^{2} e^{4} + 7 \, a c e^{6}\right )} x^{2} - 2 \, {\left (16 \, c^{2} d^{3} e^{3} + 29 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{189 \, {\left (e^{8} x + d e^{7}\right )}} \]

[In]

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

[Out]

-2/189*(8*(32*c^2*d^6 + 81*a*c*d^4*e^2 + 57*a^2*d^2*e^4 + (32*c^2*d^5*e + 81*a*c*d^3*e^3 + 57*a^2*d*e^5)*x)*sq
rt(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) + 24*(32*c^2*d^5*e + 57*a*c*d^3*e^3 + 21*a^2*d*e^5 + (32*c^2*d^4*e^2 + 57*a*c*d^2*e^4 + 21*a^2*e^6)*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*(7*c^2*e^6*x^4 - 10*c^
2*d*e^5*x^3 - 128*c^2*d^4*e^2 - 212*a*c*d^2*e^4 - 63*a^2*e^6 + 4*(4*c^2*d^2*e^4 + 7*a*c*e^6)*x^2 - 2*(16*c^2*d
^3*e^3 + 29*a*c*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(e^8*x + d*e^7)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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