\(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [668]

Optimal result

Integrand size = 21, antiderivative size = 410 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{5 e^3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {8 \sqrt {-a} c^{3/2} \left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {32 \sqrt {-a} c^{3/2} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-2/5*(c*x^2+a)^(3/2)/e/(e*x+d)^(5/2)-4/5*c*(2*d*(a*e^2+2*c*d^2)+e*(3*a*e^2+5*c*d^2)*x)*(c*x^2+a)^(1/2)/e^3/(a*
e^2+c*d^2)/(e*x+d)^(3/2)-8/5*c^(3/2)*(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))*(-a)^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^4/(a*e^2+c*d^2)/(c*x^2+
a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+32/5*c^(3/2)*d*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)*(1+c*x^2/a)^(1/2)*((e*x+d)*c^(1/2)/(e*(
-a)^(1/2)+d*c^(1/2)))^(1/2)/e^4/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 825, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {8 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {32 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {4 c \sqrt {a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{5 e^3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \]

[In]

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

[Out]

(-4*c*(2*d*(2*c*d^2 + a*e^2) + e*(5*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(5*e^3*(c*d^2 + a*e^2)*(d + e*x)^(3/2
)) - (2*(a + c*x^2)^(3/2))/(5*e*(d + e*x)^(5/2)) - (8*Sqrt[-a]*c^(3/2)*(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)])
/(5*e^4*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (32*Sqrt[-a]*c^(
3/2)*d*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)])/(5*e^4*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*} \text {integral}& = -\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {(6 c) \int \frac {x \sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx}{5 e} \\ & = -\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{5 e^3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {(4 c) \int \frac {a c d e-c \left (4 c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{5 e^3 \left (c d^2+a e^2\right )} \\ & = -\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{5 e^3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {\left (16 c^2 d\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{5 e^4}+\frac {\left (4 c^2 \left (4 c d^2+3 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{5 e^4 \left (c d^2+a e^2\right )} \\ & = -\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{5 e^3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {\left (8 a c^{3/2} \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 )}{5 \sqrt {-a} e^4 \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (32 a c^{3/2} d \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 )}{5 \sqrt {-a} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (5 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{5 e^3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {8 \sqrt {-a} c^{3/2} \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 )}{5 e^4 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {32 \sqrt {-a} c^{3/2} d \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 )}{5 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.18 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-e^2 \left (a+c x^2\right ) \left (\left (c d^2+a e^2\right )^2-4 c d \left (c d^2+a e^2\right ) (d+e x)+c \left (11 c d^2+7 a e^2\right ) (d+e x)^2\right )+\frac {4 c (d+e x)^2 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c d^2+3 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-4 i c^{3/2} d^3+4 \sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (4 c d^2+i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{5 e^5 \left (c d^2+a e^2\right ) (d+e x)^{5/2} \sqrt {a+c x^2}} \]

[In]

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

[Out]

(2*(-(e^2*(a + c*x^2)*((c*d^2 + a*e^2)^2 - 4*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2 + 7*a*e^2)*(d + e*x)^
2)) + (4*c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c*d^2 + 3*a*e^2)*(a + c*x^2) + Sqrt[c]*((-4*I)
*c^(3/2)*d^3 + 4*Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*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)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sq
rt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - 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)/S
qrt[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]]))/(5*e^5*(c*d^2 + a*e
^2)*(d + e*x)^(5/2)*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(338)=676\).

Time = 2.64 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.92

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{6} \left (x +\frac {d}{e}\right )^{3}}+\frac {8 c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{5} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+a e \right ) c \left (7 e^{2} a +11 c \,d^{2}\right )}{5 e^{4} \left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (-\frac {11 c^{2} d}{5 e^{4}}+\frac {c^{2} d \left (7 e^{2} a +11 c \,d^{2}\right )}{5 e^{4} \left (e^{2} a +c \,d^{2}\right )}\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 {c^{2}}{e^{3}}+\frac {c^{2} \left (7 e^{2} a +11 c \,d^{2}\right )}{5 e^{3} \left (e^{2} a +c \,d^{2}\right )}\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}}\)	\(788\)
default	\(\text {Expression too large to display}\)	\(3410\)

[In]

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

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/5*(a*e^2+c*d^2)/e^6*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1
/2)/(x+d/e)^3+8/5*c/e^5*d*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2-2/5*(c*e*x^2+a*e)/e^4/(a*e^2+c*d^2)*c*(7
*a*e^2+11*c*d^2)/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(-11/5*c^2*d/e^4+1/5*c^2*d/e^4*(7*a*e^2+11*c*d^2)/(a*e^2+c*d^
2))*(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*(c^2/e^3+1/5*c^2/e^3*(7*a*e^2+11*c
*d^2)/(a*e^2+c*d^2))*(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 = 0.11 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (2 \, c^{2} d^{6} + 3 \, a c d^{4} e^{2} + {\left (2 \, c^{2} d^{3} e^{3} + 3 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{2} d^{4} e^{2} + 3 \, a c d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{2} d^{5} e + 3 \, a c d^{3} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (4 \, c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + {\left (4 \, c^{2} d^{2} e^{4} + 3 \, a c e^{6}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{3} + 3 \, a c d e^{5}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{2} + 3 \, a c d^{2} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (8 \, c^{2} d^{4} e^{2} + 5 \, a c d^{2} e^{4} + a^{2} e^{6} + {\left (11 \, c^{2} d^{2} e^{4} + 7 \, a c e^{6}\right )} x^{2} + 2 \, {\left (9 \, c^{2} d^{3} e^{3} + 5 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{15 \, {\left (c d^{5} e^{5} + a d^{3} e^{7} + {\left (c d^{2} e^{8} + a e^{10}\right )} x^{3} + 3 \, {\left (c d^{3} e^{7} + a d e^{9}\right )} x^{2} + 3 \, {\left (c d^{4} e^{6} + a d^{2} e^{8}\right )} x\right )}} \]

[In]

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

[Out]

-2/15*(8*(2*c^2*d^6 + 3*a*c*d^4*e^2 + (2*c^2*d^3*e^3 + 3*a*c*d*e^5)*x^3 + 3*(2*c^2*d^4*e^2 + 3*a*c*d^2*e^4)*x^
2 + 3*(2*c^2*d^5*e + 3*a*c*d^3*e^3)*x)*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) + 12*(4*c^2*d^5*e + 3*a*c*d^3*e^3 + (4*c^2*d^2*e^4 + 3*a*c*e^6)*x^
3 + 3*(4*c^2*d^3*e^3 + 3*a*c*d*e^5)*x^2 + 3*(4*c^2*d^4*e^2 + 3*a*c*d^2*e^4)*x)*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*(8*c^2*d^4*e^2 + 5*a*c*d^2*e^4 + a^2*e^6 + (11*c^
2*d^2*e^4 + 7*a*c*e^6)*x^2 + 2*(9*c^2*d^3*e^3 + 5*a*c*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*d^5*e^5 + a*
d^3*e^7 + (c*d^2*e^8 + a*e^10)*x^3 + 3*(c*d^3*e^7 + a*d*e^9)*x^2 + 3*(c*d^4*e^6 + a*d^2*e^8)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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