\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\) [676]

Optimal result

Integrand size = 21, antiderivative size = 553 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {8 c^2 \left (d \left (32 c^2 d^4+49 a c d^2 e^2+9 a^2 e^4\right )+e \left (40 c^2 d^4+69 a c d^2 e^2+21 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {16 \sqrt {-a} c^{5/2} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} c^{5/2} d \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-4/63*c*(2*d*(a*e^2+4*c*d^2)+e*(7*a*e^2+13*c*d^2)*x)*(c*x^2+a)^(3/2)/e^3/(a*e^2+c*d^2)/(e*x+d)^(7/2)-2/9*(c*x^
2+a)^(5/2)/e/(e*x+d)^(9/2)-8/63*c^2*(d*(9*a^2*e^4+49*a*c*d^2*e^2+32*c^2*d^4)+e*(21*a^2*e^4+69*a*c*d^2*e^2+40*c
^2*d^4)*x)*(c*x^2+a)^(1/2)/e^5/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)-16/63*c^(5/2)*(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
)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^6/(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)+16/63*c^(5/2)*d*(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)*(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/(a*e^2+c*d^2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 553, 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, 825, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {16 \sqrt {-a} c^{5/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 c^2 \sqrt {a+c x^2} \left (e x \left (21 a^2 e^4+69 a c d^2 e^2+40 c^2 d^4\right )+d \left (9 a^2 e^4+49 a c d^2 e^2+32 c^2 d^4\right )\right )}{63 e^5 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}+\frac {16 \sqrt {-a} c^{5/2} d \sqrt {\frac {c x^2}{a}+1} \left (33 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {4 c \left (a+c x^2\right )^{3/2} \left (e x \left (7 a e^2+13 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{63 e^3 (d+e x)^{7/2} \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-8*c^2*(d*(32*c^2*d^4 + 49*a*c*d^2*e^2 + 9*a^2*e^4) + e*(40*c^2*d^4 + 69*a*c*d^2*e^2 + 21*a^2*e^4)*x)*Sqrt[a
+ c*x^2])/(63*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(13*c*d^2 + 7*a*e^2)*x)
*(a + c*x^2)^(3/2))/(63*e^3*(c*d^2 + a*e^2)*(d + e*x)^(7/2)) - (2*(a + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) - (
16*Sqrt[-a]*c^(5/2)*(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[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(63*e^6*(c*d^2 + a*e^2)^2*S
qrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*c^(5/2)*d*(32*c*d^2 + 33*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)])/(63*e^6*(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 825

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^
(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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}}{9 e (d+e x)^{9/2}}+\frac {(10 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e} \\ & = -\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {(4 c) \int \frac {\left (5 a c d e-c \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx}{21 e^3 \left (c d^2+a e^2\right )} \\ & = -\frac {8 c^2 \left (d \left (32 c^2 d^4+49 a c d^2 e^2+9 a^2 e^4\right )+e \left (40 c^2 d^4+69 a c d^2 e^2+21 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {(8 c) \int \frac {-4 a c^2 d e \left (2 c d^2+3 a e^2\right )+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 e^5 \left (c d^2+a e^2\right )^2} \\ & = -\frac {8 c^2 \left (d \left (32 c^2 d^4+49 a c d^2 e^2+9 a^2 e^4\right )+e \left (40 c^2 d^4+69 a c d^2 e^2+21 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {\left (8 c^3 d \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 \left (c d^2+a e^2\right )}+\frac {\left (8 c^3 \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 \left (c d^2+a e^2\right )^2} \\ & = -\frac {8 c^2 \left (d \left (32 c^2 d^4+49 a c d^2 e^2+9 a^2 e^4\right )+e \left (40 c^2 d^4+69 a c d^2 e^2+21 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {\left (16 a c^{5/2} \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 \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 (16 a c^{5/2} d \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 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {8 c^2 \left (d \left (32 c^2 d^4+49 a c d^2 e^2+9 a^2 e^4\right )+e \left (40 c^2 d^4+69 a c d^2 e^2+21 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {16 \sqrt {-a} c^{5/2} \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 \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 {16 \sqrt {-a} c^{5/2} d \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 \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 = 15.58 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 \left (-e^2 \left (a+c x^2\right ) \left (7 \left (c d^2+a e^2\right )^4-38 c d \left (c d^2+a e^2\right )^3 (d+e x)+4 c \left (c d^2+a e^2\right )^2 \left (22 c d^2+7 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right ) \left (61 c d^2+57 a e^2\right ) (d+e x)^3+c^2 \left (193 c^2 d^4+330 a c d^2 e^2+105 a^2 e^4\right ) (d+e x)^4\right )+\frac {8 c^2 (d+e x)^4 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{5/2} d^5+32 \sqrt {a} c^2 d^4 e-57 i a c^{3/2} d^3 e^2+57 a^{3/2} c d^2 e^3-21 i a^2 \sqrt {c} d e^4+21 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c^2 d^4+8 i \sqrt {a} c^{3/2} d^3 e+57 a c d^2 e^2+12 i a^{3/2} \sqrt {c} d e^3+21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{63 e^7 \left (c d^2+a e^2\right )^2 (d+e x)^{9/2} \sqrt {a+c x^2}} \]

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(2*(-(e^2*(a + c*x^2)*(7*(c*d^2 + a*e^2)^4 - 38*c*d*(c*d^2 + a*e^2)^3*(d + e*x) + 4*c*(c*d^2 + a*e^2)^2*(22*c*
d^2 + 7*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)*(61*c*d^2 + 57*a*e^2)*(d + e*x)^3 + c^2*(193*c^2*d^4 + 33
0*a*c*d^2*e^2 + 105*a^2*e^4)*(d + e*x)^4)) + (8*c^2*(d + e*x)^4*(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])/Sqr
t[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[Sqr
t[-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]*S
qrt[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)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(63*e^7*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2)*Sqrt[a + c*x^2]
)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1007\) vs. \(2(475)=950\).

Time = 5.17 (sec) , antiderivative size = 1008, normalized size of antiderivative = 1.82

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1008\)
default	\(\text {Expression too large to display}\)	\(8244\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/9*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^10*(c*e*x^3+c*
d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^5+76/63/e^9*(a*e^2+c*d^2)*c*d*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^4-8/63*
(7*a*e^2+22*c*d^2)*c/e^8*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3+4/63*c^2*d*(57*a*e^2+61*c*d^2)/(a*e^2+c*d
^2)/e^7*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2-2/63*(c*e*x^2+a*e)/e^6/(a*e^2+c*d^2)^2*c^2*(105*a^2*e^4+33
0*a*c*d^2*e^2+193*c^2*d^4)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(-5*c^3*d/e^6+2/63*c^3*d*(57*a*e^2+61*c*d^2)/(a*e^2
+c*d^2)/e^6+1/63*c^3*d/e^6*(105*a^2*e^4+330*a*c*d^2*e^2+193*c^2*d^4)/(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))+2*(c^3/e^5+1/63*c^3/e^5*(105*a^2*e^4+330*a*c*d^2*e^2+193*c^2*d^4
)/(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 = 0.14 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.02 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

-2/189*(8*(32*c^4*d^10 + 81*a*c^3*d^8*e^2 + 57*a^2*c^2*d^6*e^4 + (32*c^4*d^5*e^5 + 81*a*c^3*d^3*e^7 + 57*a^2*c
^2*d*e^9)*x^5 + 5*(32*c^4*d^6*e^4 + 81*a*c^3*d^4*e^6 + 57*a^2*c^2*d^2*e^8)*x^4 + 10*(32*c^4*d^7*e^3 + 81*a*c^3
*d^5*e^5 + 57*a^2*c^2*d^3*e^7)*x^3 + 10*(32*c^4*d^8*e^2 + 81*a*c^3*d^6*e^4 + 57*a^2*c^2*d^4*e^6)*x^2 + 5*(32*c
^4*d^9*e + 81*a*c^3*d^7*e^3 + 57*a^2*c^2*d^5*e^5)*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) + 24*(32*c^4*d^9*e + 57*a*c^3*d^7*e^3 + 21*a^2*c^2*d
^5*e^5 + (32*c^4*d^4*e^6 + 57*a*c^3*d^2*e^8 + 21*a^2*c^2*e^10)*x^5 + 5*(32*c^4*d^5*e^5 + 57*a*c^3*d^3*e^7 + 21
*a^2*c^2*d*e^9)*x^4 + 10*(32*c^4*d^6*e^4 + 57*a*c^3*d^4*e^6 + 21*a^2*c^2*d^2*e^8)*x^3 + 10*(32*c^4*d^7*e^3 + 5
7*a*c^3*d^5*e^5 + 21*a^2*c^2*d^3*e^7)*x^2 + 5*(32*c^4*d^8*e^2 + 57*a*c^3*d^6*e^4 + 21*a^2*c^2*d^4*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*(128*c^4*d^8*e^2 + 212
*a*c^3*d^6*e^4 + 63*a^2*c^2*d^4*e^6 + 18*a^3*c*d^2*e^8 + 7*a^4*e^10 + (193*c^4*d^4*e^6 + 330*a*c^3*d^2*e^8 + 1
05*a^2*c^2*e^10)*x^4 + 2*(325*c^4*d^5*e^5 + 542*a*c^3*d^3*e^7 + 153*a^2*c^2*d*e^9)*x^3 + 4*(220*c^4*d^6*e^4 +
369*a*c^3*d^4*e^6 + 108*a^2*c^2*d^2*e^8 + 7*a^3*c*e^10)*x^2 + 2*(272*c^4*d^7*e^3 + 453*a*c^3*d^5*e^5 + 126*a^2
*c^2*d^3*e^7 + 9*a^3*c*d*e^9)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c^2*d^9*e^7 + 2*a*c*d^7*e^9 + a^2*d^5*e^11 +
(c^2*d^4*e^12 + 2*a*c*d^2*e^14 + a^2*e^16)*x^5 + 5*(c^2*d^5*e^11 + 2*a*c*d^3*e^13 + a^2*d*e^15)*x^4 + 10*(c^2*
d^6*e^10 + 2*a*c*d^4*e^12 + a^2*d^2*e^14)*x^3 + 10*(c^2*d^7*e^9 + 2*a*c*d^5*e^11 + a^2*d^3*e^13)*x^2 + 5*(c^2*
d^8*e^8 + 2*a*c*d^6*e^10 + a^2*d^4*e^12)*x)

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**(11/2), x)

Maxima [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(11/2), x)

Giac [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(11/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^(11/2),x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^(11/2), x)