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

Optimal. Leaf size=393 \[ -\frac {8 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (5 a e^2+4 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 \sqrt {c} e^4 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {32 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a e^2+4 c d^2-3 c d e x\right )}{35 e^3}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 e} \]

[Out]

2/7*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e+4/35*(-3*c*d*e*x+5*a*e^2+4*c*d^2)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^3+32/35*
d*(2*a*e^2+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)-8/35*(a*e^2+c*d^2)*(5*a*e^2+4*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*x^2/a+1)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))
^(1/2)/e^4/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.36, antiderivative size = 393, 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 = {735, 815, 844, 719, 424, 419} \[ \frac {4 \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a e^2+4 c d^2-3 c d e x\right )}{35 e^3}-\frac {8 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (5 a e^2+4 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 \sqrt {c} e^4 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {32 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[d + e*x]*(4*c*d^2 + 5*a*e^2 - 3*c*d*e*x)*Sqrt[a + c*x^2])/(35*e^3) + (2*Sqrt[d + e*x]*(a + c*x^2)^(3/2
))/(7*e) + (32*Sqrt[-a]*Sqrt[c]*d*(c*d^2 + 2*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)])/(35*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt
[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2 + a*e^2)*(4*c*d^2 + 5*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)])/(35*Sqrt[c]*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 735

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 + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(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] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 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}}{\sqrt {d+e x}} \, dx &=\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 e}+\frac {6 \int \frac {(a e-c d x) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{7 e}\\ &=\frac {4 \sqrt {d+e x} \left (4 c d^2+5 a e^2-3 c d e x\right ) \sqrt {a+c x^2}}{35 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 e}+\frac {8 \int \frac {\frac {1}{2} a c e \left (c d^2+5 a e^2\right )-2 c^2 d \left (c d^2+2 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{35 c e^3}\\ &=\frac {4 \sqrt {d+e x} \left (4 c d^2+5 a e^2-3 c d e x\right ) \sqrt {a+c x^2}}{35 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 e}-\frac {\left (16 c d \left (c d^2+2 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{35 e^4}+\frac {\left (4 \left (c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{35 e^4}\\ &=\frac {4 \sqrt {d+e x} \left (4 c d^2+5 a e^2-3 c d e x\right ) \sqrt {a+c x^2}}{35 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 e}-\frac {\left (32 a \sqrt {c} d \left (c d^2+2 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 )}{35 \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 (8 a \left (c d^2+a e^2\right ) \left (4 c d^2+5 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 )}{35 \sqrt {-a} \sqrt {c} e^4 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {4 \sqrt {d+e x} \left (4 c d^2+5 a e^2-3 c d e x\right ) \sqrt {a+c x^2}}{35 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 e}+\frac {32 \sqrt {-a} \sqrt {c} d \left (c d^2+2 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 )}{35 e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (4 c d^2+5 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 )}{35 \sqrt {c} e^4 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 3.18, size = 575, normalized size = 1.46 \[ \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (15 a e^2+c \left (8 d^2-6 d e x+5 e^2 x^2\right )\right )}{e^3}-\frac {8 \left (-\sqrt {a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+i \sqrt {a} c d^2 e+8 a \sqrt {c} d e^2+4 c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+4 \sqrt {c} d (d+e x)^{3/2} \left (2 a^{3/2} e^3+\sqrt {a} c d^2 e-2 i a \sqrt {c} d e^2-i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (2 a^2 e^2+a c \left (d^2+2 e^2 x^2\right )+c^2 d^2 x^2\right )\right )}{e^5 (d+e x) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{35 \sqrt {a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(15*a*e^2 + c*(8*d^2 - 6*d*e*x + 5*e^2*x^2)))/e^3 - (8*(4*d*e^2*Sqrt[-d - (I*Sq
rt[a]*e)/Sqrt[c]]*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)) + 4*Sqrt[c]*d*((-I)*c^(3/2)*d^3 + Sqrt[a]*
c*d^2*e - (2*I)*a*Sqrt[c]*d*e^2 + 2*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]]/Sqrt[d
+ e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e +
8*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)/Sq
rt[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)]))/(e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(35*
Sqrt[a + c*x^2])

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.08, size = 1385, normalized size = 3.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^(1/2),x)

[Out]

-2/35*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(-5*c^3*e^5*x^5+20*(-a*c)^(1/2)*(-(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^5+36*(-a*
c)^(1/2)*(-(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*d^2*e^3+16*(-a*c)^(1/2)*(-(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))*c^2*d^4*e+12*a^2*c*
(-(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))*d*e^4+12*a*c^2*(-(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))*d^3*e^2-32*a^2*c*(-(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
))*d*e^4-48*a*c^2*(-(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))*d^3*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^3*d^5+c^3*d*e^4*x^4-20*a*
c^2*e^5*x^3-2*c^3*d^2*e^3*x^3-14*a*c^2*d*e^4*x^2-8*c^3*d^3*e^2*x^2-15*a^2*c*e^5*x-2*a*c^2*d^2*e^3*x-15*a^2*c*d
*e^4-8*a*c^2*d^3*e^2)/c/e^5/(c*e*x^3+c*d*x^2+a*e*x+a*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(3/2)/(d + e*x)^(1/2),x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**(1/2),x)

[Out]

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

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