3.666 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\)

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

[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 = 0.31, 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 = {733, 815, 844, 719, 424, 419} \[ \frac {32 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {8 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{5 e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {4 c \sqrt {a+c x^2} (4 d-3 e x) \sqrt {d+e x}}{5 e^3}-\frac {2 \left (a+c x^2\right )^{3/2}}{e \sqrt {d+e x}} \]

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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 719

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 733

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 815

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 844

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 ) \operatorname {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 ) \operatorname {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]  time = 2.96, size = 565, normalized size = 1.53 \[ \frac {2 \sqrt {a+c x^2} \left (c \left (-8 d^2-2 d e x+e^2 x^2\right )-5 a e^2\right )}{5 e^3 \sqrt {d+e x}}+\frac {8 \left (\sqrt {c} (d+e x)^{3/2} \left (3 a^{3/2} e^3+4 \sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2-4 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (3 a^2 e^2+a c \left (4 d^2+3 e^2 x^2\right )+4 c^2 d^2 x^2\right )-\sqrt {a} \sqrt {c} e (d+e x)^{3/2} \left (i \sqrt {a} \sqrt {c} d e+3 a e^2+4 c d^2\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{5 e^5 \sqrt {a+c x^2} \sqrt {d+e x} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}} \]

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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fricas [F]  time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]

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]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

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)/(e*x + d)^(3/2), x)

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maple [B]  time = 0.10, size = 1168, normalized size = 3.17 \[ \frac {2 \sqrt {c \,x^{2}+a}\, \sqrt {e x +d}\, \left (c^{2} e^{4} x^{4}-2 c^{2} d \,e^{3} x^{3}-4 a c \,e^{4} x^{2}-8 c^{2} d^{2} e^{2} x^{2}-12 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a^{2} e^{4} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+12 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a^{2} e^{4} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )-28 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a c \,d^{2} e^{2} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+12 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a c \,d^{2} e^{2} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )-2 a c d \,e^{3} x -16 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, c^{2} d^{4} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )-5 a^{2} e^{4}-8 a c \,d^{2} e^{2}+16 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, \sqrt {-a c}\, a d \,e^{3} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+16 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, \sqrt {-a c}\, c \,d^{3} e \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )\right )}{5 \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(12*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c
)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*
e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*e^4+12*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)
^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*E
llipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c*d^2
*e^2+16*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*
c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/
2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*(-a*c)^(1/2)*a*d*e^3+16*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a
*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticF((-(e*x+d
)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*(-a*c)^(1/2)*c*d^3*e-12*
(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2)
)/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c
*d+(-a*c)^(1/2)*e))^(1/2))*a^2*e^4-28*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c
)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*
e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c*d^2*e^2-16*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e
)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/
2)*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*c^2
*d^4+c^2*e^4*x^4-2*c^2*d*e^3*x^3-4*a*c*e^4*x^2-8*c^2*d^2*e^2*x^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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

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)/(e*x + d)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]

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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