3.7.77 \(\int \frac {(d+e x)^{7/2}}{\sqrt {a+c x^2}} \, dx\) [677]

Optimal. Leaf size=413 \[ \frac {2 e \left (71 c d^2-25 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{105 c^2}+\frac {24 d e (d+e x)^{3/2} \sqrt {a+c x^2}}{35 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {32 \sqrt {-a} d \left (11 c d^2-13 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \left (71 c d^2-25 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

24/35*d*e*(e*x+d)^(3/2)*(c*x^2+a)^(1/2)/c+2/7*e*(e*x+d)^(5/2)*(c*x^2+a)^(1/2)/c+2/105*e*(-25*a*e^2+71*c*d^2)*(
e*x+d)^(1/2)*(c*x^2+a)^(1/2)/c^2-32/105*d*(-13*a*e^2+11*c*d^2)*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)/c^(3/2)/(c*x^2+a)
^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+2/105*(-25*a*e^2+71*c*d^2)*(a*e^2+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)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.43, antiderivative size = 413, 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 = {757, 847, 858, 733, 435, 430} \begin {gather*} \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (71 c d^2-25 a e^2\right ) \left (a e^2+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 )}{105 c^{5/2} \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {32 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (11 c d^2-13 a e^2\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 )}{105 c^{3/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 e \sqrt {a+c x^2} \sqrt {d+e x} \left (71 c d^2-25 a e^2\right )}{105 c^2}+\frac {2 e \sqrt {a+c x^2} (d+e x)^{5/2}}{7 c}+\frac {24 d e \sqrt {a+c x^2} (d+e x)^{3/2}}{35 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/Sqrt[a + c*x^2],x]

[Out]

(2*e*(71*c*d^2 - 25*a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(105*c^2) + (24*d*e*(d + e*x)^(3/2)*Sqrt[a + c*x^2])
/(35*c) + (2*e*(d + e*x)^(5/2)*Sqrt[a + c*x^2])/(7*c) - (32*Sqrt[-a]*d*(11*c*d^2 - 13*a*e^2)*Sqrt[d + e*x]*Sqr
t[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)
])/(105*c^(3/2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*(71*c*d^2 -
25*a*e^2)*(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[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(105*c^(5/2)*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 757

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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 {(d+e x)^{7/2}}{\sqrt {a+c x^2}} \, dx &=\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (7 c d^2-5 a e^2\right )+6 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{7 c}\\ &=\frac {24 d e (d+e x)^{3/2} \sqrt {a+c x^2}}{35 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} c d \left (35 c d^2-61 a e^2\right )+\frac {1}{4} c e \left (71 c d^2-25 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{35 c^2}\\ &=\frac {2 e \left (71 c d^2-25 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{105 c^2}+\frac {24 d e (d+e x)^{3/2} \sqrt {a+c x^2}}{35 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {8 \int \frac {\frac {1}{8} c \left (105 c^2 d^4-254 a c d^2 e^2+25 a^2 e^4\right )+2 c^2 d e \left (11 c d^2-13 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{105 c^3}\\ &=\frac {2 e \left (71 c d^2-25 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{105 c^2}+\frac {24 d e (d+e x)^{3/2} \sqrt {a+c x^2}}{35 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {\left (16 d \left (11 c d^2-13 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{105 c}-\frac {\left (\left (71 c d^2-25 a e^2\right ) \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{105 c^2}\\ &=\frac {2 e \left (71 c d^2-25 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{105 c^2}+\frac {24 d e (d+e x)^{3/2} \sqrt {a+c x^2}}{35 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {\left (32 a d \left (11 c d^2-13 a e^2\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 )}{105 \sqrt {-a} c^{3/2} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a \left (71 c d^2-25 a e^2\right ) \left (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 )}{105 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {2 e \left (71 c d^2-25 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{105 c^2}+\frac {24 d e (d+e x)^{3/2} \sqrt {a+c x^2}}{35 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {32 \sqrt {-a} d \left (11 c d^2-13 a e^2\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 )}{105 c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \left (71 c d^2-25 a e^2\right ) \left (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 )}{105 c^{5/2} \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 = 21.98, size = 548, normalized size = 1.33 \begin {gather*} \frac {2 \sqrt {d+e x} \left (\frac {16 d e \left (-13 a^2 e^2+11 c^2 d^2 x^2+a c \left (11 d^2-13 e^2 x^2\right )\right )}{d+e x}+\left (a+c x^2\right ) \left (-25 a e^3+c e \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+\frac {16 i c d \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (11 c d^2-13 a e^2\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 )}{e}+\frac {\left (105 i c^2 d^4-176 \sqrt {a} c^{3/2} d^3 e-254 i a c d^2 e^2+208 a^{3/2} \sqrt {c} d e^3+25 i 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} 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 )}{e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{105 c^2 \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/Sqrt[a + c*x^2],x]

[Out]

(2*Sqrt[d + e*x]*((16*d*e*(-13*a^2*e^2 + 11*c^2*d^2*x^2 + a*c*(11*d^2 - 13*e^2*x^2)))/(d + e*x) + (a + c*x^2)*
(-25*a*e^3 + c*e*(122*d^2 + 66*d*e*x + 15*e^2*x^2)) + ((16*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(11*c*d^2 -
13*a*e^2)*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)])/e + (((105*I)*c^2*d^4 - 176*Sqrt[a]*c^(3/2)*d^3*e - (254*I)*a*c*d^2*e^2 + 208*a^(3/2)*S
qrt[c]*d*e^3 + (25*I)*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]*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*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(105*c^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. \(1533\) vs. \(2(335)=670\).
time = 0.50, size = 1534, normalized size = 3.71

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{7 c}+\frac {44 d \,e^{2} x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{35 c}+\frac {2 \left (\frac {122 d^{2} e^{2}}{35}-\frac {5 a \,e^{4}}{7 c}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (d^{4}-\frac {44 a \,d^{2} e^{2}}{35 c}-\frac {a \left (\frac {122 d^{2} e^{2}}{35}-\frac {5 a \,e^{4}}{7 c}\right )}{3 c}\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 (4 d^{3} e -\frac {86 a d \,e^{3}}{35 c}-\frac {2 d \left (\frac {122 d^{2} e^{2}}{35}-\frac {5 a \,e^{4}}{7 c}\right )}{3 e}\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}}\)	\(748\)
risch	\(-\frac {2 e \left (-15 x^{2} c \,e^{2}-66 c d e x +25 e^{2} a -122 c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{105 c^{2}}+\frac {\left (-\frac {2 \left (208 a c \,e^{3} d -176 c^{2} d^{3} e \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}}+\frac {50 a^{2} e^{4} \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 {508 a c \,d^{2} e^{2} \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 {210 c^{2} d^{4} \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}}\right ) \sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}}{105 c^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(1067\)
default	\(\text {Expression too large to display}\)	\(1534\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/105*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(-15*c^3*e^5*x^5+25*(-(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^2*e^5-46*(-(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*d^2*e^3-71*(-(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^2*d^4*e+183*(-(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*d*e^4+78*(-(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^2*d^3*e^2-105*(-(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))*c^3*d^5-208*(-(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*d*e^4-32*(-(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^2*d^3*e^2+176*(-(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))*c^3*d^5-81*c^3*d*e^
4*x^4+10*a*c^2*e^5*x^3-188*c^3*d^2*e^3*x^3-56*a*c^2*d*e^4*x^2-122*c^3*d^3*e^2*x^2+25*a^2*c*e^5*x-188*a*c^2*d^2
*e^3*x+25*d*e^4*a^2*c-122*d^3*e^2*c^2*a)/e/(c*e*x^3+c*d*x^2+a*e*x+a*d)/c^3

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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((e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((x*e + d)^(7/2)/sqrt(c*x^2 + a), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.76, size = 256, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left ({\left (139 \, c^{2} d^{4} - 554 \, a c d^{2} e^{2} + 75 \, a^{2} e^{4}\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 ) - 48 \, {\left (11 \, c^{2} d^{3} e - 13 \, a c d 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 (66 \, c^{2} d x e^{3} + 122 \, c^{2} d^{2} e^{2} + 5 \, {\left (3 \, c^{2} x^{2} - 5 \, a c\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{315 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

2/315*((139*c^2*d^4 - 554*a*c*d^2*e^2 + 75*a^2*e^4)*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)) - 48*(11*c^2*d^3*e - 13*a*c*d*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, weierstrassPInv
erse(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*(66*c^2*
d*x*e^3 + 122*c^2*d^2*e^2 + 5*(3*c^2*x^2 - 5*a*c)*e^4)*sqrt(c*x^2 + a)*sqrt(x*e + d))*e^(-1)/c^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\sqrt {a + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(c*x**2+a)**(1/2),x)

[Out]

Integral((d + e*x)**(7/2)/sqrt(a + c*x**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((e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate((x*e + d)^(7/2)/sqrt(c*x^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {c\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(a + c*x^2)^(1/2),x)

[Out]

int((d + e*x)^(7/2)/(a + c*x^2)^(1/2), x)

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