3.7.94 \(\int \frac {(d+e x)^{5/2}}{(a+c x^2)^{5/2}} \, dx\) [694]

Optimal. Leaf size=392 \[ -\frac {(a e-c d x) (d+e x)^{3/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 c \sqrt {a+c x^2}}+\frac {\left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 d \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-1/3*(-c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(c*x^2+a)^(3/2)-1/6*(a*d*e-(3*a*e^2+4*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c/(c*x^2
+a)^(1/2)+1/6*(3*a*e^2+4*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))*(e*x+d)^(1/2)*(c*x^2/a+1)^(1/2)/(-a)^(3/2)/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)-2/3*d*(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))*(c*x^2/a+1)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/(-a)^(3/2)/
c^(3/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {753, 835, 858, 733, 435, 430} \begin {gather*} -\frac {\sqrt {d+e x} \left (a d e-x \left (3 a e^2+4 c d^2\right )\right )}{6 a^2 c \sqrt {a+c x^2}}-\frac {2 d \sqrt {\frac {c x^2}{a}+1} \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 )}{3 (-a)^{3/2} c^{3/2} \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 a e^2+4 c d^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 )}{6 (-a)^{3/2} c^{3/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {(d+e x)^{3/2} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((a*e - c*d*x)*(d + e*x)^(3/2))/(a*c*(a + c*x^2)^(3/2)) - (Sqrt[d + e*x]*(a*d*e - (4*c*d^2 + 3*a*e^2)*x))
/(6*a^2*c*Sqrt[a + c*x^2]) + ((4*c*d^2 + 3*a*e^2)*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)])/(6*(-a)^(3/2)*c^(3/2)*Sqrt[(Sqrt[c]*(d +
e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (2*d*(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[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]
*Sqrt[c]*d - a*e)])/(3*(-a)^(3/2)*c^(3/2)*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 753

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

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(
a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*
x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x
] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 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 {(d+e x)^{5/2}}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {(a e-c d x) (d+e x)^{3/2}}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (4 c d^2+3 a e^2\right )+\frac {1}{2} c d e x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 c \sqrt {a+c x^2}}+\frac {\int \frac {\frac {1}{4} a c d e^2-\frac {1}{4} c e \left (4 c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 c \sqrt {a+c x^2}}+\frac {\left (d \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 c}-\frac {\left (4 c d^2+3 a e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{12 a^2 c}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 c \sqrt {a+c x^2}}-\frac {\left (\left (4 c d^2+3 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 )}{6 \sqrt {-a} 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 d \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 )}{3 \sqrt {-a} a c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {(a e-c d x) (d+e x)^{3/2}}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a d e-\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 c \sqrt {a+c x^2}}+\frac {\left (4 c d^2+3 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 )}{6 (-a)^{3/2} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 d \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 )}{3 (-a)^{3/2} c^{3/2} \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 = 22.21, size = 597, normalized size = 1.52 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {2 \left (4 c^2 d^2 x^3+a^2 e (-3 d+e x)+a c x \left (6 d^2+d e x+3 e^2 x^2\right )\right )}{a^2 c \left (a+c x^2\right )}+\frac {(d+e x) \left (-\frac {2 e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (3 a^2 e^2+4 c^2 d^2 x^2+a c \left (4 d^2+3 e^2 x^2\right )\right )}{(d+e x)^2}+\frac {2 i \sqrt {c} \left (4 c^{3/2} d^3+4 i \sqrt {a} c d^2 e+3 a \sqrt {c} d e^2+3 i a^{3/2} e^3\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}} 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 {d+e x}}+\frac {2 \sqrt {a} \sqrt {c} e \left (4 c d^2+i \sqrt {a} \sqrt {c} d e+3 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}} 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+e x}}\right )}{a^2 c^2 e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{12 \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(4*c^2*d^2*x^3 + a^2*e*(-3*d + e*x) + a*c*x*(6*d^2 + d*e*x + 3*e^2*x^2)))/(a^2*c*(a + c*x^2
)) + ((d + e*x)*((-2*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)
))/(d + e*x)^2 + ((2*I)*Sqrt[c]*(4*c^(3/2)*d^3 + (4*I)*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + (3*I)*a^(3/2)*e^3
)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticE[I*A
rcSinh[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 + e*x] + (2*Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*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))]*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 + e*x]))/(a^2*c^2*e*Sqrt
[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(12*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2277\) vs. \(2(320)=640\).
time = 0.50, size = 2278, normalized size = 5.81

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{3 c^{3} a}-\frac {2 d e}{3 c^{3}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (-\frac {\left (3 e^{2} a +4 c \,d^{2}\right ) x}{12 a^{2} c^{2}}-\frac {d e}{12 a \,c^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {2 d \left (e^{2} a +c \,d^{2}\right )}{3 c \,a^{2}}-\frac {d \,e^{2}}{12 c a}-\frac {d \left (3 e^{2} a +4 c \,d^{2}\right )}{6 c \,a^{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 {e \left (3 e^{2} a +4 c \,d^{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 )}{6 a^{2} c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(755\)
default	\(\text {Expression too large to display}\)	\(2278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(-(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*e^4*x^2+7*(-(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^2*e^2*x^2+4*(-(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^4*x^2-4*(-(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*e^3*x^2-4*(-(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^
3*e*x^2-3*(-(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*e^4*x^2-3*(-(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^2*e^2*x^2+3*(-(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*e^4+7*(-(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^2*e^2+4*(-(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^4-4*(-(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*d*e^3-4*(-(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^3*e-3*(-(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
pticF((-(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*e^4-3*(-
(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^2*e^2+3*a*c^2*e^4*x^4+4*c^3*d^2*e^2*x^4+4*a*c^2*d*e^3*x^3+4*c^3*d^3*e*x^3+a^2*c*e
^4*x^2+7*a*c^2*d^2*e^2*x^2-2*a^2*c*d*e^3*x+6*a*c^2*d^3*e*x-3*d^2*e^2*a^2*c)/c^2/(e*x+d)^(1/2)/a^2/e/(c*x^2+a)^
(3/2)

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.24, size = 394, normalized size = 1.01 \begin {gather*} \frac {{\left (2 \, {\left (2 \, c^{3} d^{3} x^{4} + 4 \, a c^{2} d^{3} x^{2} + 2 \, a^{2} c d^{3} + 3 \, {\left (a c^{2} d x^{4} + 2 \, a^{2} c d x^{2} + a^{3} d\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 ) + 3 \, {\left (3 \, {\left (a c^{2} x^{4} + 2 \, a^{2} c x^{2} + a^{3}\right )} e^{3} + 4 \, {\left (c^{3} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e\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 \, \sqrt {c x^{2} + a} {\left ({\left (3 \, a c^{2} x^{3} + a^{2} c x\right )} e^{3} + {\left (a c^{2} d x^{2} - 3 \, a^{2} c d\right )} e^{2} + 2 \, {\left (2 \, c^{3} d^{2} x^{3} + 3 \, a c^{2} d^{2} x\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{18 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(2*(2*c^3*d^3*x^4 + 4*a*c^2*d^3*x^2 + 2*a^2*c*d^3 + 3*(a*c^2*d*x^4 + 2*a^2*c*d*x^2 + a^3*d)*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)) + 3*(3*(a*c^2*x^4 + 2*a^2*c*x^2 + a^3)*e^3 + 4*(c^3*d^2*x^4 + 2*a*c^2*d^2*x^2 + a^2*c*d^2)*e)*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, weierstrassPInve
rse(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*sqrt(c*x^
2 + a)*((3*a*c^2*x^3 + a^2*c*x)*e^3 + (a*c^2*d*x^2 - 3*a^2*c*d)*e^2 + 2*(2*c^3*d^2*x^3 + 3*a*c^2*d^2*x)*e)*sqr
t(x*e + d))*e^(-1)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

integrate((x*e + d)^(5/2)/(c*x^2 + a)^(5/2), 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 )}^{5/2}}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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