3.7.66 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [666]

Optimal. Leaf size=369 \[ -\frac {4 c (4 d-3 e x) \sqrt {d+e x} \sqrt {a+c x^2}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}}-\frac {8 \sqrt {-a} \sqrt {c} \left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {32 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 369, 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 = {747, 829, 858, 733, 435, 430} \begin {gather*} \frac {32 \sqrt {-a} \sqrt {c} 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 )}{5 e^4 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {8 \sqrt {-a} \sqrt {c} \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 )}{5 e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {4 c \sqrt {a+c x^2} (4 d-3 e x) \sqrt {d+e x}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c*(4*d - 3*e*x)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(5*e^3) - (2*(a + c*x^2)^(3/2))/(e*Sqrt[d + e*x]) - (8*Sqrt
[-a]*Sqrt[c]*(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*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]
*Sqrt[a + c*x^2]) + (32*Sqrt[-a]*Sqrt[c]*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)])/(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 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 )^{3/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {(6 c) \int \frac {x \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {4 c (4 d-3 e x) \sqrt {d+e x} \sqrt {a+c x^2}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {8 \int \frac {-\frac {1}{2} a c d e+\frac {1}{2} 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}\\ &=-\frac {4 c (4 d-3 e x) \sqrt {d+e x} \sqrt {a+c x^2}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}}-\frac {\left (16 c d \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{5 e^4}+\frac {\left (4 c \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}\\ &=-\frac {4 c (4 d-3 e x) \sqrt {d+e x} \sqrt {a+c x^2}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {\left (8 a \sqrt {c} \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 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (32 a \sqrt {c} 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 )}{5 \sqrt {-a} e^4 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {4 c (4 d-3 e x) \sqrt {d+e x} \sqrt {a+c x^2}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}}-\frac {8 \sqrt {-a} \sqrt {c} \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 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {32 \sqrt {-a} \sqrt {c} 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 )}{5 e^4 \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 = 21.90, size = 565, normalized size = 1.53 \begin {gather*} \frac {2 \sqrt {a+c x^2} \left (-5 a e^2+c \left (-8 d^2-2 d e x+e^2 x^2\right )\right )}{5 e^3 \sqrt {d+e x}}+\frac {8 \left (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 )+\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 \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 {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} 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 )\right )}{5 e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {d+e x} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + c*x^2]*(-5*a*e^2 + c*(-8*d^2 - 2*d*e*x + e^2*x^2)))/(5*e^3*Sqrt[d + e*x]) + (8*(e^2*Sqrt[-d - (I*S
qrt[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)) + Sqrt[c]*((-4*I)*c^(3/2)*d^3 + 4*Sqr
t[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*Sqrt[a]*e)/Sqrt[c]]/Sq
rt[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]*Sq
rt[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))]*(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)]))/(5*e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1167\) vs. \(2(299)=598\).
time = 0.54, size = 1168, normalized size = 3.17

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right ) \left (e^{2} a +c \,d^{2}\right )}{e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{2}}-\frac {6 d c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{3}}+\frac {2 \left (-\frac {c d \left (2 e^{2} a +c \,d^{2}\right )}{e^{4}}+\frac {\left (e^{2} a +c \,d^{2}\right ) c d}{e^{4}}+\frac {d a c}{5 e^{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 {2 \left (\frac {c \left (2 e^{2} a +c \,d^{2}\right )}{e^{3}}+\frac {\left (e^{2} a +c \,d^{2}\right ) c}{e^{3}}-\frac {3 a c}{5 e}+\frac {6 d^{2} c^{2}}{5 e^{3}}\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}}\)	\(755\)
default	\(\frac {2 \sqrt {c \,x^{2}+a}\, \sqrt {e x +d}\, \left (16 \sqrt {-a c}\, \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a d \,e^{3}+16 \sqrt {-a c}\, \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c \,d^{3} e +12 a^{2} \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) e^{4}+12 \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a c \,d^{2} e^{2}-12 a^{2} \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) e^{4}-28 \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a c \,d^{2} e^{2}-16 \sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c^{2} d^{4}+c^{2} e^{4} x^{4}-2 c^{2} d \,x^{3} e^{3}-4 a c \,e^{4} x^{2}-8 d^{2} e^{2} x^{2} c^{2}-2 a c d \,e^{3} x -5 a^{2} e^{4}-8 a c \,d^{2} e^{2}\right )}{5 \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) e^{5}}\)	\(1168\)
risch	\(\text {Expression too large to display}\)	\(1454\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(16*(-a*c)^(1/2)*(-(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*d*e^3+16*(-a*c)^(1/2)*(-(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*d^3*e+12*a^2*(-(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))*e^4+12*(-(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*d^2*e^2-12*a^2*(-(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))*e^4-28*(-(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*d^2*e^2-16*(-(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^2*d^4+c^2*e^4*x^4-2*c^2*d*x^3
*e^3-4*a*c*e^4*x^2-8*d^2*e^2*x^2*c^2-2*a*c*d*e^3*x-5*a^2*e^4-8*a*c*d^2*e^2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)/e^5

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.51, size = 273, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (8 \, {\left (2 \, c d^{3} x e + 2 \, c d^{4} + 3 \, a d x e^{3} + 3 \, a d^{2} 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 ) + 12 \, {\left (4 \, c d^{2} x e^{2} + 4 \, c d^{3} e + 3 \, a x e^{4} + 3 \, a 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 (2 \, c d x e^{3} + 8 \, c d^{2} e^{2} - {\left (c x^{2} - 5 \, a\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{15 \, {\left (x e^{6} + d e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c*x^2 + a)^(3/2)/(x*e + d)^(3/2), 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 )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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