3.624 \(\int (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx\)
Optimal. Leaf size=434 \[ \frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (5 a e g^2+c f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 \sqrt {c} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{105 c g^2}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c} \]
[Out]
2/7*e*(c*x^2+a)^(3/2)*(g*x+f)^(1/2)/c-2/105*(5*a*e*g^2+c*f*(-7*d*g+4*e*f)-3*c*g*(7*d*g+e*f)*x)*(g*x+f)^(1/2)*(
c*x^2+a)^(1/2)/c/g^2-4/105*(c*f^2*(-7*d*g+4*e*f)+a*g^2*(21*d*g+8*e*f))*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)/g^3/c^(1/
2)/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)+4/105*(a*g^2+c*f^2)*(5*a*e*g^2+c*f*(-7*d*g
+4*e*f))*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 = 0.49, antiderivative size = 434, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used =
{833, 815, 844, 719, 424, 419} \[ \frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (5 a e g^2+c f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 \sqrt {c} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right )}{105 c g^2}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {f+g x}}{7 c} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]
[Out]
(-2*Sqrt[f + g*x]*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g) - 3*c*g*(e*f + 7*d*g)*x)*Sqrt[a + c*x^2])/(105*c*g^2) + (2*
e*Sqrt[f + g*x]*(a + c*x^2)^(3/2))/(7*c) - (4*Sqrt[-a]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*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*Sqrt[c]*g^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) + (4*Sqrt[-
a]*(c*f^2 + a*g^2)*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g))*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^(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 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 833
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
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 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2} \, dx &=\frac {2 e \sqrt {f+g x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {2 \int \frac {\left (\frac {1}{2} (7 c d f-a e g)+\frac {1}{2} c (e f+7 d g) x\right ) \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx}{7 c}\\ &=-\frac {2 \sqrt {f+g x} \left (5 a e g^2+c f (4 e f-7 d g)-3 c g (e f+7 d g) x\right ) \sqrt {a+c x^2}}{105 c g^2}+\frac {2 e \sqrt {f+g x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {8 \int \frac {-\frac {1}{4} a c g \left (5 a e g^2+c f (e f-28 d g)\right )+\frac {1}{4} c^2 \left (c f^2 (4 e f-7 d g)+a g^2 (8 e f+21 d g)\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c^2 g^2}\\ &=-\frac {2 \sqrt {f+g x} \left (5 a e g^2+c f (4 e f-7 d g)-3 c g (e f+7 d g) x\right ) \sqrt {a+c x^2}}{105 c g^2}+\frac {2 e \sqrt {f+g x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {\left (2 \left (c f^2+a g^2\right ) \left (5 a e g^2+c f (4 e f-7 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c g^3}+\frac {\left (2 \left (c f^2 (4 e f-7 d g)+a g^2 (8 e f+21 d g)\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{105 g^3}\\ &=-\frac {2 \sqrt {f+g x} \left (5 a e g^2+c f (4 e f-7 d g)-3 c g (e f+7 d g) x\right ) \sqrt {a+c x^2}}{105 c g^2}+\frac {2 e \sqrt {f+g x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {\left (4 a \left (c f^2 (4 e f-7 d g)+a g^2 (8 e f+21 d g)\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} \sqrt {c} g^3 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (4 a \left (c f^2+a g^2\right ) \left (5 a e g^2+c f (4 e f-7 d g)\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^{3/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {2 \sqrt {f+g x} \left (5 a e g^2+c f (4 e f-7 d g)-3 c g (e f+7 d g) x\right ) \sqrt {a+c x^2}}{105 c g^2}+\frac {2 e \sqrt {f+g x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {4 \sqrt {-a} \left (c f^2 (4 e f-7 d g)+a g^2 (8 e f+21 d g)\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 \sqrt {c} g^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (5 a e g^2+c f (4 e f-7 d g)\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^{3/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 4.58, size = 610, normalized size = 1.41 \[ \frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (10 a e g^2+7 c d g (f+3 g x)+c e \left (-4 f^2+3 f g x+15 g^2 x^2\right )\right )}{c g^2}+\frac {4 \left (g^2 \left (a+c x^2\right ) \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right )+i \sqrt {c} (f+g x)^{3/2} \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} \left (c f^2 (7 d g-4 e f)-a g^2 (21 d g+8 e f)\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} g (f+g x)^{3/2} \left (-\sqrt {a} g+i \sqrt {c} f\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} \left (3 \sqrt {a} \sqrt {c} g (7 d g+e f)+5 i a e g^2+i c f (4 e f-7 d g)\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^4 (f+g x) \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{105 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]
[Out]
(Sqrt[f + g*x]*((2*(a + c*x^2)*(10*a*e*g^2 + 7*c*d*g*(f + 3*g*x) + c*e*(-4*f^2 + 3*f*g*x + 15*g^2*x^2)))/(c*g^
2) + (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*(a + c*x^2) + I
*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(c*f^2*(-4*e*f + 7*d*g) - a*g^2*(8*e*f + 21*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt
[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[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[a]*g*
(I*Sqrt[c]*f - Sqrt[a]*g)*((5*I)*a*e*g^2 + I*c*f*(4*e*f - 7*d*g) + 3*Sqrt[a]*Sqrt[c]*g*(e*f + 7*d*g))*Sqrt[(g*
((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*Ellipti
cF[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)]))/(c*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*Sqrt[a + c*x^2])
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + a} {\left (e x + d\right )} \sqrt {g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + a} {\left (e x + d\right )} \sqrt {g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f), x)
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maple [B] time = 0.03, size = 2549, normalized size = 5.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)*(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)*(4*a*c^2*e*f^3*g^2-10*a^2*c*e*f*g^4-7*a*c^2*d*f^2*g^3-21*x^4*c^3*d*g^5-15
*x^5*c^3*e*g^5-18*x^4*c^3*e*f*g^4+14*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c
)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f)
)^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*a*c*d*f*g^4-18*(-(g*x+f)*c/((-a*c)^(1
/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f
))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))
*(-a*c)^(1/2)*a*c*e*f^2*g^3+8*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)
*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)
,(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^3*e*f^5+24*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c
*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(
g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*e*f^3*g^2+14*(-
(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g
/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a
*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c^2*d*f^3*g^2-8*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))
*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)
^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-a*c)^(1/2)*c^2*e*f^4*g+16*(-(g*x+f)
*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c
)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/
2)*g))^(1/2))*a^2*c*e*f*g^4+28*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2
)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2
),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*d*f^2*g^3-6*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2
)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*Ellipti
cF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a^2*c*e*f*g^4-4
2*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2
))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f
+(-a*c)^(1/2)*g))^(1/2))*a*c^2*d*f^2*g^3-6*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f
+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*
g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a*c^2*e*f^3*g^2-10*(-(g*x+f)*c/((-a*c)^(1/2)
*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^
(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*(-
a*c)^(1/2)*a^2*e*g^5-25*x^3*a*c^2*e*g^5-28*x^3*c^3*d*f*g^4+x^3*c^3*e*f^2*g^3-21*x^2*a*c^2*d*g^5-7*x^2*c^3*d*f^
2*g^3+4*x^2*c^3*e*f^3*g^2-10*x*a^2*c*e*g^5+x*a*c^2*e*f^2*g^3-28*x^2*a*c^2*e*f*g^4-28*x*a*c^2*d*f*g^4+42*(-(g*x
+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-
a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^
(1/2)*g))^(1/2))*a^2*c*d*g^5-14*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/
2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/
2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*c^3*d*f^4*g-42*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*
((-c*x+(-a*c)^(1/2))*g/(c*f+(-a*c)^(1/2)*g))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF
((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/(c*f+(-a*c)^(1/2)*g))^(1/2))*a^2*c*d*g^5)/(c*g
*x^3+c*f*x^2+a*g*x+a*f)/g^4/c^2
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + a} {\left (e x + d\right )} \sqrt {g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)^(1/2)*(a + c*x^2)^(1/2)*(d + e*x),x)
[Out]
int((f + g*x)^(1/2)*(a + c*x^2)^(1/2)*(d + e*x), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + c x^{2}} \left (d + e x\right ) \sqrt {f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(g*x+f)**(1/2)*(c*x**2+a)**(1/2),x)
[Out]
Integral(sqrt(a + c*x**2)*(d + e*x)*sqrt(f + g*x), x)
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