3.7.71 \(\int \frac {(a+c x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [671]

Optimal. Leaf size=494 \[ \frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {16 \sqrt {-a} \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

20/693*(-7*c*d*e*x+9*a*e^2+8*c*d^2)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e^3+2/11*(c*x^2+a)^(5/2)*(e*x+d)^(1/2)/e+8/6
93*(32*c^2*d^4+69*a*c*d^2*e^2+45*a^2*e^4-24*c*d*e*(2*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5+16/693*
d*(93*a^2*e^4+93*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)*(c*x^2/a+1)^(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)-16/693*(a*e^2+c*d^2)*(45*a^2*e^4+69*a*c*d^2*e^2+32*c^2*d^4)*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/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 494, 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 = {749, 829, 858, 733, 435, 430} \begin {gather*} -\frac {16 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\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 )}{693 \sqrt {c} e^6 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (93 a^2 e^4+93 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 )}{693 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {8 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{693 e^5}+\frac {20 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{693 e^3}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[d + e*x]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4 - 24*c*d*e*(c*d^2 + 2*a*e^2)*x)*Sqrt[a + c*x^2])/(6
93*e^5) + (20*Sqrt[d + e*x]*(8*c*d^2 + 9*a*e^2 - 7*c*d*e*x)*(a + c*x^2)^(3/2))/(693*e^3) + (2*Sqrt[d + e*x]*(a
+ c*x^2)^(5/2))/(11*e) + (16*Sqrt[-a]*Sqrt[c]*d*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*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)]
)/(693*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*(c*d^2 + a*e^2)*
(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4)*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)])/(693*Sqrt[
c]*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 749

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 + 2*p + 1))), x] + Dist[2*(p/(e*(m + 2*p + 1))), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(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] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 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*} \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx &=\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {10 \int \frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 e}\\ &=\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {40 \int \frac {\left (\frac {1}{2} a c e \left (c d^2+9 a e^2\right )-4 c^2 d \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{231 c e^3}\\ &=\frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {32 \int \frac {\frac {1}{4} a c^2 e \left (8 c^2 d^4+21 a c d^2 e^2+45 a^2 e^4\right )-\frac {1}{4} c^3 d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{693 c^2 e^5}\\ &=\frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {\left (8 \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{693 e^6}-\frac {\left (8 c d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{693 e^6}\\ &=\frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}-\frac {\left (16 a \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 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 )}{693 \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 \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\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 )}{693 \sqrt {-a} \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {16 \sqrt {-a} \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 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 )}{693 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\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 )}{693 \sqrt {c} e^6 \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 = 22.45, size = 634, normalized size = 1.28 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-\frac {8 d e^2 \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \left (a+c x^2\right )}{d+e x}+e^2 \left (a+c x^2\right ) \left (333 a^2 e^4+2 a c e^2 \left (178 d^2-131 d e x+108 e^2 x^2\right )+c^2 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )-8 i c d \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+93 a c d^2 e^2+93 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}} \sqrt {d+e x} 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 )+\frac {8 \sqrt {a} e \left (32 c^{5/2} d^5+8 i \sqrt {a} c^2 d^4 e+93 a c^{3/2} d^3 e^2+21 i a^{3/2} c d^2 e^3+93 a^2 \sqrt {c} d e^4+45 i 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}} \sqrt {d+e x} 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 )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{693 e^7 \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*((-8*d*e^2*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*(a + c*x^2))/(d + e*x) + e^2*(a + c*x^2
)*(333*a^2*e^4 + 2*a*c*e^2*(178*d^2 - 131*d*e*x + 108*e^2*x^2) + c^2*(128*d^4 - 96*d^3*e*x + 80*d^2*e^2*x^2 -
70*d*e^3*x^3 + 63*e^4*x^4)) - (8*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*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))]*Sqrt[d +
e*x]*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)] + (8*Sqrt[a]*e*(32*c^(5/2)*d^5 + (8*I)*Sqrt[a]*c^2*d^4*e + 93*a*c^(3/2)*d^3*e^2 + (21*I)*a^(3
/2)*c*d^2*e^3 + 93*a^2*Sqrt[c]*d*e^4 + (45*I)*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))]*Sqrt[d + e*x]*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]]))/(693
*e^7*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1969\) vs. \(2(416)=832\).
time = 0.50, size = 1970, normalized size = 3.99 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/693*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(-744*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*
e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d*e^6-1488*(-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1
/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*
c^2*d^3*e^4-1000*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1
/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^3*d^5*e^2+256*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c
/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*c^3*d^6*e+384*(-
(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e
/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^
(1/2)*e+c*d))^(1/2))*a^3*c*d*e^6+576*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1
/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d)
)^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^3*e^4+192*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*
d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)
*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^3*d
^5*e^2-63*c^4*e^7*x^7+360*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))
^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-(
(-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a^3*e^7+912*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*Ellip
ticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*
a^2*c*d^2*e^5+808*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*(
(c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(
1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a*c^2*d^4*e^3-128*d^5*e^2*c^3*a-356*d^3*e^4*a^2*c^2-333*
d*e^6*c*a^3-333*a^3*c*e^7*x+7*c^4*d*e^6*x^6-279*a*c^3*e^7*x^5-10*c^4*d^2*e^5*x^5+16*c^4*d^3*e^4*x^4-549*a^2*c^
2*e^7*x^3-32*c^4*d^4*e^3*x^3-128*c^4*d^5*e^2*x^2-104*a*c^3*d^2*e^5*x^3-256*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(
1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*Elli
pticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^4*d^7-287*
a^2*c^2*d*e^6*x^2-340*a*c^3*d^3*e^4*x^2-94*a^2*c^2*d^2*e^5*x-32*a*c^3*d^4*e^3*x+53*a*c^3*d*e^6*x^4)/c/e^7/(c*e
*x^3+c*d*x^2+a*e*x+a*d)

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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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.57, size = 343, normalized size = 0.69 \begin {gather*} \frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{6} + 117 \, a c^{2} d^{4} e^{2} + 156 \, a^{2} c d^{2} e^{4} + 135 \, a^{3} e^{6}\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 (32 \, c^{3} d^{5} e + 93 \, a c^{2} d^{3} e^{3} + 93 \, a^{2} c d e^{5}\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 (96 \, c^{3} d^{3} x e^{3} - 128 \, c^{3} d^{4} e^{2} - 9 \, {\left (7 \, c^{3} x^{4} + 24 \, a c^{2} x^{2} + 37 \, a^{2} c\right )} e^{6} + 2 \, {\left (35 \, c^{3} d x^{3} + 131 \, a c^{2} d x\right )} e^{5} - 4 \, {\left (20 \, c^{3} d^{2} x^{2} + 89 \, a c^{2} d^{2}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{2079 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/2079*(8*(32*c^3*d^6 + 117*a*c^2*d^4*e^2 + 156*a^2*c*d^2*e^4 + 135*a^3*e^6)*sqrt(c)*e^(1/2)*weierstrassPInver
se(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*(32*c^3*d^
5*e + 93*a*c^2*d^3*e^3 + 93*a^2*c*d*e^5)*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*(96*c^3*d^3*x*e^3 - 128*c^3*d^4*e^2 - 9*(7*c^3*x^4 + 24*a*c^2*x^2 + 37*a
^2*c)*e^6 + 2*(35*c^3*d*x^3 + 131*a*c^2*d*x)*e^5 - 4*(20*c^3*d^2*x^2 + 89*a*c^2*d^2)*e^4)*sqrt(c*x^2 + a)*sqrt
(x*e + d))*e^(-7)/c

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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}}}{\sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Integral((a + c*x**2)**(5/2)/sqrt(d + e*x), 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)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)/sqrt(x*e + d), 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}}{\sqrt {d+e\,x}} \,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)^(1/2),x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^(1/2), x)

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