3.645 \(\int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx\)
Optimal. Leaf size=457 \[ \frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (a e^2 g^2 (7 e f-15 d g)-c \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{15 c g^2}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g} \]
[Out]
-8/15*e^2*(-3*d*g+e*f)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g^2+2/5*e^2*(e*x+d)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g+2
/15*e*(9*a*e^2*g^2-c*(45*d^2*g^2-30*d*e*f*g+8*e^2*f^2))*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)-2/15*(a*e^2*g^2*(-15*d*g+7*e*f)-c*(-15*d^3*g^3+45*d^2*e*f
*g^2-30*d*e^2*f^2*g+8*e^3*f^3))*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^(3/2)/g^3/(
g*x+f)^(1/2)/(c*x^2+a)^(1/2)
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Rubi [A] time = 0.61, antiderivative size = 457, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used =
{931, 1654, 844, 719, 424, 419} \[ -\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (a e^2 g^2 (7 e f-15 d g)-c \left (45 d^2 e f g^2-15 d^3 g^3-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (9 a e^2 g^2-c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {8 e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-3 d g)}{15 c g^2}+\frac {2 e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}{5 c g} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^3/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
[Out]
(-8*e^2*(e*f - 3*d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(15*c*g^2) + (2*e^2*(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^
2])/(5*c*g) + (2*Sqrt[-a]*e*(9*a*e^2*g^2 - c*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*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)])/(15*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]*(a*e^2*g^2*(7*e*f
- 15*d*g) - c*(8*e^3*f^3 - 30*d*e^2*f^2*g + 45*d^2*e*f*g^2 - 15*d^3*g^3))*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)])/(15*c^(3/2)*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 931
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[(2*e^2*(
d + e*x)^(m - 2)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*g*(2*m - 1)), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(
m - 3)*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m
- 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(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] && GeQ[m, 2]
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^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c g}-\frac {\int \frac {-5 c d^3 g+a e^2 (2 e f+d g)+e \left (3 a e^2 g+c d (2 e f-15 d g)\right ) x+4 c e^2 (e f-3 d g) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{5 c g}\\ &=-\frac {8 e^2 (e f-3 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c g}-\frac {2 \int \frac {-\frac {1}{2} c g^2 \left (15 c d^3 g-a e^2 (2 e f+15 d g)\right )+\frac {1}{2} c e g \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{15 c^2 g^3}\\ &=-\frac {8 e^2 (e f-3 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c g}-\frac {\left (e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{15 c g^3}+\frac {\left (a e^2 g^2 (7 e f-15 d g)-c \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{15 c g^3}\\ &=-\frac {8 e^2 (e f-3 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c g}-\frac {\left (2 a e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\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 )}{15 \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 (2 a \left (a e^2 g^2 (7 e f-15 d g)-c \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\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 )}{15 \sqrt {-a} c^{3/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {8 e^2 (e f-3 d g) \sqrt {f+g x} \sqrt {a+c x^2}}{15 c g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}{5 c g}+\frac {2 \sqrt {-a} e \left (9 a e^2 g^2-c \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\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 )}{15 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} \left (a e^2 g^2 (7 e f-15 d g)-c \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )\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 )}{15 c^{3/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 4.40, size = 625, normalized size = 1.37 \[ \frac {2 \sqrt {f+g x} \left (\frac {\sqrt {c} g \sqrt {f+g x} \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} \left (9 a^{3/2} e^3 g^2+\sqrt {a} c e \left (-45 d^2 g^2+30 d e f g-8 e^2 f^2\right )-i a \sqrt {c} e^2 g (15 d g+2 e f)+15 i c^{3/2} d^3 g^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}+\frac {e g^2 \left (-9 a^2 e^2 g^2+a c \left (45 d^2 g^2-30 d e f g+e^2 \left (8 f^2-9 g^2 x^2\right )\right )+c^2 x^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right )}{f+g x}+i c e \sqrt {f+g x} \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} \left (c \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )-9 a e^2 g^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+c e^2 g^2 \left (a+c x^2\right ) (15 d g-4 e f+3 e g x)\right )}{15 c^2 g^4 \sqrt {a+c x^2}} \]
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]*(c*e^2*g^2*(-4*e*f + 15*d*g + 3*e*g*x)*(a + c*x^2) + (e*g^2*(-9*a^2*e^2*g^2 + c^2*(8*e^2*f^2
- 30*d*e*f*g + 45*d^2*g^2)*x^2 + a*c*(-30*d*e*f*g + 45*d^2*g^2 + e^2*(8*f^2 - 9*g^2*x^2))))/(f + g*x) + I*c*e*
Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-9*a*e^2*g^2 + c*(8*e^2*f^2 - 30*d*e*f*g + 45*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]*EllipticE[I*ArcSinh[S
qrt[-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[c
]*g*((15*I)*c^(3/2)*d^3*g^2 + 9*a^(3/2)*e^3*g^2 - I*a*Sqrt[c]*e^2*g*(2*e*f + 15*d*g) + Sqrt[a]*c*e*(-8*e^2*f^2
+ 30*d*e*f*g - 45*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]]))/(15*c^2*g^4*Sqrt[a + c*x^2])
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fricas [F] time = 1.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}}{c g x^{3} + c f x^{2} + a g x + a f}, x\right ) \]
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(c*x^2 + a)*sqrt(g*x + f)/(c*g*x^3 + c*f*x^2 + a*g*x +
a*f), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + a} \sqrt {g x + f}}\,{d x} \]
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(c*x^2 + a)*sqrt(g*x + f)), x)
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maple [B] time = 0.06, size = 2949, normalized size = 6.45 \[ \text {output too large to display} \]
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]
-2/15*(4*a*c*e^3*f^2*g^2-45*(-(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*d^2*e*f*g^3+30*(-(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*d*e^2*f^2*g^2+45*(-(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*d*e^2*f*g^3-30*(-(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
*d*e^2*f*g^3-3*x^4*c^2*e^3*g^4-15*x^2*c^2*d*e^2*f*g^3+15*(-(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*d^3*g^4+45*
(-(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^2*d^2*e*f^2*g^2-30*(-(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^2*d*e^2*f^3*g-15*x*a*c*d*e^2*g^4+
x*a*c*e^3*f*g^3-15*(-(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*d*e^2*g^4+7*(-(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)*Elli
pticF((-(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*e^3*f*g^3-8*(-(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*e^3*f^3*g-45*(-(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)*Ellip
ticF((-(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*d^2*e*g
^4-6*(-(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*e^3*f^2*g^2+45*(-(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*d^2*e*g^4-(-(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*e^3*f^2*g^2-15*(-(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^2*d^3*f*g^3-15*x^3*c^2*d*e^2*g^4+x^3*c^2*e^3*f*g^3-3*
x^2*a*c*e^3*g^4+4*x^2*c^2*e^3*f^2*g^2+9*(-(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*e^3*g^4-9*(-(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*e^3*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^2*e^3*f^4-15*a*c*d*e^2*f*g^3)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c^2/g^4/(
c*g*x^3+c*f*x^2+a*g*x+a*f)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + a} \sqrt {g x + f}}\,{d x} \]
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(c*x^2 + a)*sqrt(g*x + f)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^3/((f + g*x)^(1/2)*(a + c*x^2)^(1/2)),x)
[Out]
int((d + e*x)^3/((f + g*x)^(1/2)*(a + c*x^2)^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{3}}{\sqrt {a + c x^{2}} \sqrt {f + g x}}\, dx \]
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(a + c*x**2)*sqrt(f + g*x)), x)
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