\(\int \frac {(d+e x)^{7/2}}{(a+c x^2)^{3/2}} \, dx\) [685]

Optimal result

Integrand size = 21, antiderivative size = 426 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {d \left (3 c d^2-29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-(-c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(c*x^2+a)^(1/2)-d*e*(e*x+d)^(3/2)*(c*x^2+a)^(1/2)/a/c-1/3*e*(-5*a*e^2+3*c*d^2)
*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/a/c^2-1/3*d*(-29*a*e^2+3*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)*(1+c*x^2/a)^(1/2)/c^(3/2)/(-a)^(1/2)/(c*x^2+a)
^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+1/3*(-5*a*e^2+3*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))*(1+c*x^2/a)^(1/2)*((e*x+d)*c^(
1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/c^(5/2)/(-a)^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 426, 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 = {753, 847, 858, 733, 435, 430} \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{5/2} \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {e \sqrt {a+c x^2} \sqrt {d+e x} \left (3 c d^2-5 a e^2\right )}{3 a c^2}-\frac {(d+e x)^{5/2} (a e-c d x)}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {a+c x^2} (d+e x)^{3/2}}{a c} \]

[In]

Int[(d + e*x)^(7/2)/(a + c*x^2)^(3/2),x]

[Out]

-(((a*e - c*d*x)*(d + e*x)^(5/2))/(a*c*Sqrt[a + c*x^2])) - (e*(3*c*d^2 - 5*a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2
])/(3*a*c^2) - (d*e*(d + e*x)^(3/2)*Sqrt[a + c*x^2])/(a*c) - (d*(3*c*d^2 - 29*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)])/(3*Sq
rt[-a]*c^(3/2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + ((3*c*d^2 - 5*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[ArcSin[Sqrt[1 - (
Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*Sqrt[-a]*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 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 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*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {5 a e^2}{2}-\frac {5}{2} c d e x\right )}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}+\frac {2 \int \frac {\sqrt {d+e x} \left (10 a c d e^2-\frac {5}{4} c e \left (3 c d^2-5 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{5 a c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}+\frac {4 \int \frac {\frac {5}{8} a c e^2 \left (27 c d^2-5 a e^2\right )-\frac {5}{8} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 a c^3} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {\left (d \left (3 c d^2-29 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{6 a c}+\frac {\left (\left (3 c d^2-5 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}{6 a c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {\left (d \left (3 c d^2-29 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 )}{3 \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 (\left (3 c d^2-5 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 )}{3 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {d \left (3 c d^2-29 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 )}{3 \sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\left (3 c d^2-5 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 )}{3 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.70 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {10 a e^3}{c^2}+\frac {6 d^3 x}{a}+\frac {2 e \left (-9 d^2-9 d e x+2 e^2 x^2\right )}{c}+\frac {2 \left (-d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (3 c d^2-29 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} d \left (3 i c^{3/2} d^3-3 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+29 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (3 c^{3/2} d^3+27 i \sqrt {a} c d^2 e-29 a \sqrt {c} d e^2-5 i a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{a c^2 e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{6 \sqrt {a+c x^2}} \]

[In]

Integrate[(d + e*x)^(7/2)/(a + c*x^2)^(3/2),x]

[Out]

(Sqrt[d + e*x]*((10*a*e^3)/c^2 + (6*d^3*x)/a + (2*e*(-9*d^2 - 9*d*e*x + 2*e^2*x^2))/c + (2*(-(d*e^2*Sqrt[-d -
(I*Sqrt[a]*e)/Sqrt[c]]*(3*c*d^2 - 29*a*e^2)*(a + c*x^2)) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e -
(29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/S
qrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*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)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c*d^2*e - 29
*a*Sqrt[c]*d*e^2 - (5*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))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (S
qrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(a*c^2*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(6
*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(793\) vs. \(2(352)=704\).

Time = 5.94 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) d x}{2 c^{2} a}-\frac {e \left (e^{2} a -3 c \,d^{2}\right )}{2 c^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 e^{3} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c^{2}}+\frac {2 \left (-\frac {e^{2} \left (e^{2} a -6 c \,d^{2}\right )}{c^{2}}+\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}}{c^{2} a}-\frac {e^{2} \left (e^{2} a -3 c \,d^{2}\right )}{2 c^{2}}+\frac {d^{2} \left (3 e^{2} a -c \,d^{2}\right )}{c a}-\frac {a \,e^{4}}{3 c^{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}}}\, F\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 {10 d \,e^{3}}{3 c}+\frac {\left (3 e^{2} a -c \,d^{2}\right ) d e}{2 a 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}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\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}\, F\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}}\)	\(794\)
default	\(\text {Expression too large to display}\)	\(1362\)
risch	\(\text {Expression too large to display}\)	\(1487\)

[In]

int((e*x+d)^(7/2)/(c*x^2+a)^(3/2),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*(c*e*x+c*d)*(1/2*(3*a*e^2-c*d^2)/c^2*d/a*x-1/2*e*(
a*e^2-3*c*d^2)/c^3)/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)+2/3*e^3/c^2*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(-e^2*(a*e
^2-6*c*d^2)/c^2+1/c^2*(a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)/a-1/2/c^2*e^2*(a*e^2-3*c*d^2)+1/c*d^2*(3*a*e^2-c*d^2)/a-
1/3/c^2*a*e^4)*(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*(10/3*d*e^3/c+1/2*(3*a*
e^2-c*d^2)*d*e/a/c)*(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 = 0.18 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, a c^{2} d^{4} + 52 \, a^{2} c d^{2} e^{2} - 15 \, a^{3} e^{4} + {\left (3 \, c^{3} d^{4} + 52 \, a c^{2} d^{2} e^{2} - 15 \, a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (3 \, a c^{2} d^{3} e - 29 \, a^{2} c d e^{3} + {\left (3 \, c^{3} d^{3} e - 29 \, a c^{2} d e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (2 \, a c^{2} e^{4} x^{2} - 9 \, a c^{2} d^{2} e^{2} + 5 \, a^{2} c e^{4} + 3 \, {\left (c^{3} d^{3} e - 3 \, a c^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{9 \, {\left (a c^{4} e x^{2} + a^{2} c^{3} e\right )}} \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

1/9*((3*a*c^2*d^4 + 52*a^2*c*d^2*e^2 - 15*a^3*e^4 + (3*c^3*d^4 + 52*a*c^2*d^2*e^2 - 15*a^2*c*e^4)*x^2)*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) +
3*(3*a*c^2*d^3*e - 29*a^2*c*d*e^3 + (3*c^3*d^3*e - 29*a*c^2*d*e^3)*x^2)*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*(2*a*c^2*e^4*x^2 - 9*a*c^2*d^2*e^2 + 5*a^2*c*e^4 + 3*(c^
3*d^3*e - 3*a*c^2*d*e^3)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(a*c^4*e*x^2 + a^2*c^3*e)

Sympy [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((e*x+d)**(7/2)/(c*x**2+a)**(3/2),x)

[Out]

Integral((d + e*x)**(7/2)/(a + c*x**2)**(3/2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + a)^(3/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]

[In]

int((d + e*x)^(7/2)/(a + c*x^2)^(3/2),x)

[Out]

int((d + e*x)^(7/2)/(a + c*x^2)^(3/2), x)