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

Optimal. Leaf size=553 \[ -\frac {8 c^2 \left (d \left (32 c^2 d^4+49 a c d^2 e^2+9 a^2 e^4\right )+e \left (40 c^2 d^4+69 a c d^2 e^2+21 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (13 c d^2+7 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{63 e^3 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {16 \sqrt {-a} c^{5/2} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} c^{5/2} d \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]

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

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Rubi [A]
time = 0.41, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 825, 858, 733, 435, 430} \begin {gather*} -\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 (\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 )}{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}} 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 )}{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}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains complex when optimal does not.
time = 23.35, size = 762, normalized size = 1.38 \begin {gather*} \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 \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} \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} 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 )}{\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}} \end {gather*}

Antiderivative was successfully verified.

[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]
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8243\) vs. \(2(475)=950\).
time = 0.49, size = 8244, normalized size = 14.91

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{9 e^{10} \left (x +\frac {d}{e}\right )^{5}}+\frac {76 \left (e^{2} a +c \,d^{2}\right ) c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{63 e^{9} \left (x +\frac {d}{e}\right )^{4}}-\frac {8 \left (7 e^{2} a +22 c \,d^{2}\right ) c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{63 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c^{2} d \left (57 e^{2} a +61 c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{63 \left (e^{2} a +c \,d^{2}\right ) e^{7} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+a e \right ) c^{2} \left (105 a^{2} e^{4}+330 a c \,d^{2} e^{2}+193 c^{2} d^{4}\right )}{63 e^{6} \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 {5 c^{3} d}{e^{6}}+\frac {2 c^{3} d \left (57 e^{2} a +61 c \,d^{2}\right )}{63 \left (e^{2} a +c \,d^{2}\right ) e^{6}}+\frac {c^{3} d \left (105 a^{2} e^{4}+330 a c \,d^{2} e^{2}+193 c^{2} d^{4}\right )}{63 e^{6} \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}}}\, \EllipticF \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 {2 \left (\frac {c^{3}}{e^{5}}+\frac {c^{3} \left (105 a^{2} e^{4}+330 a c \,d^{2} e^{2}+193 c^{2} d^{4}\right )}{63 e^{5} \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}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \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}\, \EllipticF \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 )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(1008\)
default	\(\text {Expression too large to display}\)	\(8244\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.68, size = 1082, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left (8 \, {\left (160 \, c^{4} d^{9} x e + 32 \, c^{4} d^{10} + 57 \, a^{2} c^{2} d x^{5} e^{9} + 285 \, a^{2} c^{2} d^{2} x^{4} e^{8} + 3 \, {\left (27 \, a c^{3} d^{3} x^{5} + 190 \, a^{2} c^{2} d^{3} x^{3}\right )} e^{7} + 15 \, {\left (27 \, a c^{3} d^{4} x^{4} + 38 \, a^{2} c^{2} d^{4} x^{2}\right )} e^{6} + {\left (32 \, c^{4} d^{5} x^{5} + 810 \, a c^{3} d^{5} x^{3} + 285 \, a^{2} c^{2} d^{5} x\right )} e^{5} + {\left (160 \, c^{4} d^{6} x^{4} + 810 \, a c^{3} d^{6} x^{2} + 57 \, a^{2} c^{2} d^{6}\right )} e^{4} + 5 \, {\left (64 \, c^{4} d^{7} x^{3} + 81 \, a c^{3} d^{7} x\right )} e^{3} + {\left (320 \, c^{4} d^{8} x^{2} + 81 \, a c^{3} d^{8}\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 ) + 24 \, {\left (160 \, c^{4} d^{8} x e^{2} + 32 \, c^{4} d^{9} e + 21 \, a^{2} c^{2} x^{5} e^{10} + 105 \, a^{2} c^{2} d x^{4} e^{9} + 3 \, {\left (19 \, a c^{3} d^{2} x^{5} + 70 \, a^{2} c^{2} d^{2} x^{3}\right )} e^{8} + 15 \, {\left (19 \, a c^{3} d^{3} x^{4} + 14 \, a^{2} c^{2} d^{3} x^{2}\right )} e^{7} + {\left (32 \, c^{4} d^{4} x^{5} + 570 \, a c^{3} d^{4} x^{3} + 105 \, a^{2} c^{2} d^{4} x\right )} e^{6} + {\left (160 \, c^{4} d^{5} x^{4} + 570 \, a c^{3} d^{5} x^{2} + 21 \, a^{2} c^{2} d^{5}\right )} e^{5} + 5 \, {\left (64 \, c^{4} d^{6} x^{3} + 57 \, a c^{3} d^{6} x\right )} e^{4} + {\left (320 \, c^{4} d^{7} x^{2} + 57 \, a c^{3} d^{7}\right )} e^{3}\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 (544 \, c^{4} d^{7} x e^{3} + 128 \, c^{4} d^{8} e^{2} + 7 \, {\left (15 \, a^{2} c^{2} x^{4} + 4 \, a^{3} c x^{2} + a^{4}\right )} e^{10} + 18 \, {\left (17 \, a^{2} c^{2} d x^{3} + a^{3} c d x\right )} e^{9} + 6 \, {\left (55 \, a c^{3} d^{2} x^{4} + 72 \, a^{2} c^{2} d^{2} x^{2} + 3 \, a^{3} c d^{2}\right )} e^{8} + 4 \, {\left (271 \, a c^{3} d^{3} x^{3} + 63 \, a^{2} c^{2} d^{3} x\right )} e^{7} + {\left (193 \, c^{4} d^{4} x^{4} + 1476 \, a c^{3} d^{4} x^{2} + 63 \, a^{2} c^{2} d^{4}\right )} e^{6} + 2 \, {\left (325 \, c^{4} d^{5} x^{3} + 453 \, a c^{3} d^{5} x\right )} e^{5} + 4 \, {\left (220 \, c^{4} d^{6} x^{2} + 53 \, a c^{3} d^{6}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{189 \, {\left (5 \, c^{2} d^{8} x e^{8} + c^{2} d^{9} e^{7} + a^{2} x^{5} e^{16} + 5 \, a^{2} d x^{4} e^{15} + 2 \, {\left (a c d^{2} x^{5} + 5 \, a^{2} d^{2} x^{3}\right )} e^{14} + 10 \, {\left (a c d^{3} x^{4} + a^{2} d^{3} x^{2}\right )} e^{13} + {\left (c^{2} d^{4} x^{5} + 20 \, a c d^{4} x^{3} + 5 \, a^{2} d^{4} x\right )} e^{12} + {\left (5 \, c^{2} d^{5} x^{4} + 20 \, a c d^{5} x^{2} + a^{2} d^{5}\right )} e^{11} + 10 \, {\left (c^{2} d^{6} x^{3} + a c d^{6} x\right )} e^{10} + 2 \, {\left (5 \, c^{2} d^{7} x^{2} + a c d^{7}\right )} e^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/189*(8*(160*c^4*d^9*x*e + 32*c^4*d^10 + 57*a^2*c^2*d*x^5*e^9 + 285*a^2*c^2*d^2*x^4*e^8 + 3*(27*a*c^3*d^3*x^
5 + 190*a^2*c^2*d^3*x^3)*e^7 + 15*(27*a*c^3*d^4*x^4 + 38*a^2*c^2*d^4*x^2)*e^6 + (32*c^4*d^5*x^5 + 810*a*c^3*d^
5*x^3 + 285*a^2*c^2*d^5*x)*e^5 + (160*c^4*d^6*x^4 + 810*a*c^3*d^6*x^2 + 57*a^2*c^2*d^6)*e^4 + 5*(64*c^4*d^7*x^
3 + 81*a*c^3*d^7*x)*e^3 + (320*c^4*d^8*x^2 + 81*a*c^3*d^8)*e^2)*sqrt(c)*e^(1/2)*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)) + 24*(160*c^4*d^8*x*e^2 + 32
*c^4*d^9*e + 21*a^2*c^2*x^5*e^10 + 105*a^2*c^2*d*x^4*e^9 + 3*(19*a*c^3*d^2*x^5 + 70*a^2*c^2*d^2*x^3)*e^8 + 15*
(19*a*c^3*d^3*x^4 + 14*a^2*c^2*d^3*x^2)*e^7 + (32*c^4*d^4*x^5 + 570*a*c^3*d^4*x^3 + 105*a^2*c^2*d^4*x)*e^6 + (
160*c^4*d^5*x^4 + 570*a*c^3*d^5*x^2 + 21*a^2*c^2*d^5)*e^5 + 5*(64*c^4*d^6*x^3 + 57*a*c^3*d^6*x)*e^4 + (320*c^4
*d^7*x^2 + 57*a*c^3*d^7)*e^3)*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*(544*c^4*d^7*x*e^3 + 128*c^4*d^8*e^2 + 7*(15*a^2*c^2*x^4 + 4*a^3*c*x^2 + a^4)*e^10
+ 18*(17*a^2*c^2*d*x^3 + a^3*c*d*x)*e^9 + 6*(55*a*c^3*d^2*x^4 + 72*a^2*c^2*d^2*x^2 + 3*a^3*c*d^2)*e^8 + 4*(271
*a*c^3*d^3*x^3 + 63*a^2*c^2*d^3*x)*e^7 + (193*c^4*d^4*x^4 + 1476*a*c^3*d^4*x^2 + 63*a^2*c^2*d^4)*e^6 + 2*(325*
c^4*d^5*x^3 + 453*a*c^3*d^5*x)*e^5 + 4*(220*c^4*d^6*x^2 + 53*a*c^3*d^6)*e^4)*sqrt(c*x^2 + a)*sqrt(x*e + d))/(5
*c^2*d^8*x*e^8 + c^2*d^9*e^7 + a^2*x^5*e^16 + 5*a^2*d*x^4*e^15 + 2*(a*c*d^2*x^5 + 5*a^2*d^2*x^3)*e^14 + 10*(a*
c*d^3*x^4 + a^2*d^3*x^2)*e^13 + (c^2*d^4*x^5 + 20*a*c*d^4*x^3 + 5*a^2*d^4*x)*e^12 + (5*c^2*d^5*x^4 + 20*a*c*d^
5*x^2 + a^2*d^5)*e^11 + 10*(c^2*d^6*x^3 + a*c*d^6*x)*e^10 + 2*(5*c^2*d^7*x^2 + a*c*d^7)*e^9)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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