3.636 \(\int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\)

Optimal. Leaf size=531 \[ -\frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} \left (25 a e^2 g^2+c \left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )}{105 c^2 g^2}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{35 c g^2}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c} \]

[Out]

2/35*e^2*(11*d*g+e*f)*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/c/g^2-2/105*e*(25*a*e^2*g^2+c*(-90*d^2*g^2+12*d*e*f*g+7*e^
2*f^2))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c^2/g^2+2/7*e*(e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c+2/105*(a*e^2*g^2
*(189*d*g+19*e*f)-c*(105*d^3*g^3+105*d^2*e*f*g^2-42*d*e^2*f^2*g+8*e^3*f^3))*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1
/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(g*x+f)^(1/2)*(c*x^2/a+1)^(1/2)/c^(3
/2)/g^3/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-2/105*e*(a*g^2+c*f^2)*(25*a*e^2*g^2-c
*(105*d^2*g^2-42*d*e*f*g+8*e^2*f^2))*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)
^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(c*x^2/a+1)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/c^(5/2)/
g^3/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 1.12, antiderivative size = 527, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {942, 1654, 844, 719, 424, 419} \[ -\frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^2 e f g^2+105 d^3 g^3-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} \left (-e^2 \left (\frac {25 a}{c}+\frac {7 f^2}{g^2}\right )+90 d^2-\frac {12 d e f}{g}\right )}{105 c}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{35 c g^2}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(90*d^2 - e^2*((25*a)/c + (7*f^2)/g^2) - (12*d*e*f)/g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(105*c) + (2*e*(d +
e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*c) + (2*e^2*(e*f + 11*d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(35*c*g^
2) + (2*Sqrt[-a]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3
))*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[
-a]*Sqrt[c]*f - a*g)])/(105*c^(3/2)*g^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) -
(2*Sqrt[-a]*e*(c*f^2 + a*g^2)*(25*a*e^2*g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2))*Sqrt[(Sqrt[c]*(f + g*x
))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2
*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(105*c^(5/2)*g^3*Sqrt[f + g*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 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]

Rule 942

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(2*e*(d
+ e*x)^(m - 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*(2*m + 1)), x] - Dist[1/(c*(2*m + 1)), Int[((d + e*x)^(m - 2)
*Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)))*x -
c*e*(e*f + d*g*(4*m - 1))*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 1]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx &=\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}-\frac {\int \frac {(d+e x) \left (-7 c d^2 f+a e (4 e f+d g)+\left (5 a e^2 g-c d (12 e f+7 d g)\right ) x-c e (e f+11 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{7 c}\\ &=\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {2 \int \frac {-\frac {1}{2} c g^2 \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e f g+5 d^2 g^2\right )\right )+\frac {1}{2} c g \left (a e^2 g^2 (23 e f+63 d g)+c \left (2 e^3 f^3+22 d e^2 f^2 g-95 d^2 e f g^2-35 d^3 g^3\right )\right ) x+\frac {1}{2} c e g^2 \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{35 c^2 g^3}\\ &=-\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {4 \int \frac {-\frac {1}{4} c g^4 \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )+\frac {1}{4} c^2 g^3 \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c^3 g^5}\\ &=-\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {\left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{105 c g^3}+-\frac {\left (4 \left (-\frac {1}{4} c g^5 \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )-\frac {1}{4} c^2 f g^3 \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c^3 g^6}\\ &=-\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {\left (2 a \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{105 \sqrt {-a} c^{3/2} g^3 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+-\frac {\left (8 a \left (-\frac {1}{4} c g^5 \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )-\frac {1}{4} c^2 f g^3 \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right )\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\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} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{105 \sqrt {-a} c^{7/2} g^6 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}+\frac {2 \sqrt {-a} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {f+g 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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} e \left (c f^2+a g^2\right ) \left (8 c e^2 f^2-42 c d e f g+105 c d^2 g^2-25 a e^2 g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 6.16, size = 747, normalized size = 1.41 \[ \frac {2 \sqrt {f+g x} \left (\frac {g \sqrt {f+g x} \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} \left (25 a^{3/2} e^3 g^2+\sqrt {a} c e \left (-105 d^2 g^2+42 d e f g-8 e^2 f^2\right )+3 i a \sqrt {c} e^2 g (2 e f-63 d g)+105 i c^{3/2} d^3 g^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}+\frac {g^2 \left (-a^2 e^2 g^2 (189 d g+19 e f)+a c \left (105 d^3 g^3+105 d^2 e f g^2-21 d e^2 g \left (2 f^2+9 g^2 x^2\right )+e^3 \left (8 f^3-19 f g^2 x^2\right )\right )+c^2 x^2 \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )}{f+g x}-\left (g^2 \left (a+c x^2\right ) \left (25 a e^3 g^2+c e \left (-105 d^2 g^2-21 d e g (f+3 g x)+e^2 \left (4 f^2-3 f g x-15 g^2 x^2\right )\right )\right )\right )+i c \sqrt {f+g x} \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} \left (c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )-a e^2 g^2 (189 d g+19 e f)\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{105 c^2 g^4 \sqrt {a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(-(g^2*(a + c*x^2)*(25*a*e^3*g^2 + c*e*(-105*d^2*g^2 - 21*d*e*g*(f + 3*g*x) + e^2*(4*f^2 - 3*
f*g*x - 15*g^2*x^2)))) + (g^2*(-(a^2*e^2*g^2*(19*e*f + 189*d*g)) + c^2*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e
*f*g^2 + 105*d^3*g^3)*x^2 + a*c*(105*d^2*e*f*g^2 + 105*d^3*g^3 - 21*d*e^2*g*(2*f^2 + 9*g^2*x^2) + e^3*(8*f^3 -
19*f*g^2*x^2))))/(f + g*x) + I*c*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-(a*e^2*g^2*(19*e*f + 189*d*g)) + c*(8*e^3
*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-((
(I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sq
rt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (g*(Sqrt[c]*f + I*Sqrt[a]*g)*((105*I)*c^(
3/2)*d^3*g^2 + 25*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(2*e*f - 63*d*g) + Sqrt[a]*c*e*(-8*e^2*f^2 + 42*d*e*
f*g - 105*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*
x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]
*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(105*c^2*g^4*Sqrt[a + c*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 3924, normalized size = 7.39 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(-25*a^2*c*e^3*f*g^4-4*a*c^2*e^3*f^3*g^2+15*x^5*c^3*e^3*g^5+84*x*a*c^2*d*e
^2*f*g^4-8*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(
-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^
(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^3*e^3*f^5-42*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1
/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*
f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*a*c*d*e^2*f*g^4-1
89*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1
/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)
/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*d*e^2*f^2*g^3+105*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(
1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c
*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*a*c*d^2*e*g^5-17
*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2
))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(
c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*a*c*e^3*f^2*g^3-105*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(
-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x
+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*d^2*e*f*g^4+231*
(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2)
)/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c
*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*d*e^2*f^2*g^3+105*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2
))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+
(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c^2*d^2*e*f^2*g^3-4
2*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/
2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/
(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c^2*d*e^2*f^3*g^2-105*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*
x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(
g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*d^3*g^5+6*(-(
g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(
-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+
(-a*c)^(1/2)*g))^(1/2))*a*c^2*e^3*f^3*g^2+8*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f
+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^
(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c^2*e^3*f^4*g+11*(-(g*x+f)
/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(
-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)
^(1/2)*g))^(1/2))*a*c^2*e^3*f^3*g^2-105*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a
*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2
)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^3*d^2*e*f^3*g^2+42*(-(g*x+f)/(-c*f+(-a*c)
^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g
)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1
/2))*c^3*d*e^2*f^4*g+19*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^
(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-
(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a^2*c*e^3*f*g^4+189*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2
)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*Ellipt
icE((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a^2*c*d*e^2*
g^5-189*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*
c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/
2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a^2*c*d*e^2*g^5+6*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1
/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*
f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a^2*c*e^3*f*g^4+63*x^4*c^3*d*e
^2*g^5+63*x^2*a*c^2*d*e^2*g^5-7*x^2*a*c^2*e^3*f*g^4+105*x^2*c^3*d^2*e*f*g^4+21*x^2*c^3*d*e^2*f^2*g^3+105*x*a*c
^2*d^2*e*g^5-x*a*c^2*e^3*f^2*g^3+84*x^3*c^3*d*e^2*f*g^4-105*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-
a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticE((-(g*x+
f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^3*d^3*f^2*g^3+105*(-(
g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(
-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+
(-a*c)^(1/2)*g))^(1/2))*a*c^2*d^3*g^5-25*(-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-
a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/
2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*a^2*e^3*g^5+105*(-(g*x+f)/(-c
*f+(-a*c)^(1/2)*g)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*f+(-a*c)^(1/2)*g)*g)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*f+(-a*c
)^(1/2)*g)*g)^(1/2)*EllipticF((-(g*x+f)/(-c*f+(-a*c)^(1/2)*g)*c)^(1/2),(-(-c*f+(-a*c)^(1/2)*g)/(c*f+(-a*c)^(1/
2)*g))^(1/2))*c^3*d^3*f^2*g^3+21*a*c^2*d*e^2*f^2*g^3+105*a*c^2*d^2*e*f*g^4+18*x^4*c^3*e^3*f*g^4-10*x^3*a*c^2*e
^3*g^5+105*x^3*c^3*d^2*e*g^5-x^3*c^3*e^3*f^2*g^3-4*x^2*c^3*e^3*f^3*g^2-25*x*a^2*c*e^3*g^5)/(c*g*x^3+c*f*x^2+a*
g*x+a*f)/c^3/g^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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