3.1.58 \(\int \frac {(a+b x^2)^3}{(c+d x^2) \sqrt {e+f x^2}} \, dx\) [58]

Optimal. Leaf size=304 \[ -\frac {b^2 (b c-a d) x \sqrt {e+f x^2}}{2 d^2 f}-\frac {3 b^2 (b e-2 a f) x \sqrt {e+f x^2}}{8 d f^2}+\frac {b^2 x \left (a+b x^2\right ) \sqrt {e+f x^2}}{4 d f}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d^3 \sqrt {d e-c f}}+\frac {b (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d^3 \sqrt {f}}+\frac {b (b c-a d) (b e-2 a f) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{2 d^2 f^{3/2}}+\frac {b \left (3 b^2 e^2-8 a b e f+8 a^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{8 d f^{5/2}} \]

[Out]

1/2*b*(-a*d+b*c)*(-2*a*f+b*e)*arctanh(x*f^(1/2)/(f*x^2+e)^(1/2))/d^2/f^(3/2)+1/8*b*(8*a^2*f^2-8*a*b*e*f+3*b^2*
e^2)*arctanh(x*f^(1/2)/(f*x^2+e)^(1/2))/d/f^(5/2)+b*(-a*d+b*c)^2*arctanh(x*f^(1/2)/(f*x^2+e)^(1/2))/d^3/f^(1/2
)-(-a*d+b*c)^3*arctan(x*(-c*f+d*e)^(1/2)/c^(1/2)/(f*x^2+e)^(1/2))/d^3/c^(1/2)/(-c*f+d*e)^(1/2)-1/2*b^2*(-a*d+b
*c)*x*(f*x^2+e)^(1/2)/d^2/f-3/8*b^2*(-2*a*f+b*e)*x*(f*x^2+e)^(1/2)/d/f^2+1/4*b^2*x*(b*x^2+a)*(f*x^2+e)^(1/2)/d
/f

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {559, 427, 396, 223, 212, 537, 385, 211} \begin {gather*} \frac {b \left (8 a^2 f^2-8 a b e f+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{8 d f^{5/2}}-\frac {(b c-a d)^3 \text {ArcTan}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d^3 \sqrt {d e-c f}}-\frac {b^2 x \sqrt {e+f x^2} (b c-a d)}{2 d^2 f}-\frac {3 b^2 x \sqrt {e+f x^2} (b e-2 a f)}{8 d f^2}+\frac {b^2 x \left (a+b x^2\right ) \sqrt {e+f x^2}}{4 d f}+\frac {b (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d^3 \sqrt {f}}+\frac {b (b c-a d) (b e-2 a f) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{2 d^2 f^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b^2*(b*c - a*d)*x*Sqrt[e + f*x^2])/(d^2*f) - (3*b^2*(b*e - 2*a*f)*x*Sqrt[e + f*x^2])/(8*d*f^2) + (b^2*x*
(a + b*x^2)*Sqrt[e + f*x^2])/(4*d*f) - ((b*c - a*d)^3*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(
Sqrt[c]*d^3*Sqrt[d*e - c*f]) + (b*(b*c - a*d)^2*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(d^3*Sqrt[f]) + (b*(b*c
- a*d)*(b*e - 2*a*f)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(2*d^2*f^(3/2)) + (b*(3*b^2*e^2 - 8*a*b*e*f + 8*a^2
*f^2)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2]])/(8*d*f^(5/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 559

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/b, Int[
(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Dist[(b*c - a*d)/b, Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*
x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx &=\frac {b \int \frac {\left (a+b x^2\right )^2}{\sqrt {e+f x^2}} \, dx}{d}+\frac {(-b c+a d) \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx}{d}\\ &=\frac {b^2 x \left (a+b x^2\right ) \sqrt {e+f x^2}}{4 d f}-\frac {(b (b c-a d)) \int \frac {a+b x^2}{\sqrt {e+f x^2}} \, dx}{d^2}+\frac {(b c-a d)^2 \int \frac {a+b x^2}{\left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx}{d^2}+\frac {b \int \frac {-a (b e-4 a f)-3 b (b e-2 a f) x^2}{\sqrt {e+f x^2}} \, dx}{4 d f}\\ &=-\frac {b^2 (b c-a d) x \sqrt {e+f x^2}}{2 d^2 f}-\frac {3 b^2 (b e-2 a f) x \sqrt {e+f x^2}}{8 d f^2}+\frac {b^2 x \left (a+b x^2\right ) \sqrt {e+f x^2}}{4 d f}+\frac {\left (b (b c-a d)^2\right ) \int \frac {1}{\sqrt {e+f x^2}} \, dx}{d^3}-\frac {(b c-a d)^3 \int \frac {1}{\left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx}{d^3}+\frac {(b (b c-a d) (b e-2 a f)) \int \frac {1}{\sqrt {e+f x^2}} \, dx}{2 d^2 f}+\frac {\left (b \left (3 b^2 e^2-8 a b e f+8 a^2 f^2\right )\right ) \int \frac {1}{\sqrt {e+f x^2}} \, dx}{8 d f^2}\\ &=-\frac {b^2 (b c-a d) x \sqrt {e+f x^2}}{2 d^2 f}-\frac {3 b^2 (b e-2 a f) x \sqrt {e+f x^2}}{8 d f^2}+\frac {b^2 x \left (a+b x^2\right ) \sqrt {e+f x^2}}{4 d f}+\frac {\left (b (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{1-f x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{d^3}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{c-(-d e+c f) x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{d^3}+\frac {(b (b c-a d) (b e-2 a f)) \text {Subst}\left (\int \frac {1}{1-f x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{2 d^2 f}+\frac {\left (b \left (3 b^2 e^2-8 a b e f+8 a^2 f^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-f x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{8 d f^2}\\ &=-\frac {b^2 (b c-a d) x \sqrt {e+f x^2}}{2 d^2 f}-\frac {3 b^2 (b e-2 a f) x \sqrt {e+f x^2}}{8 d f^2}+\frac {b^2 x \left (a+b x^2\right ) \sqrt {e+f x^2}}{4 d f}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d^3 \sqrt {d e-c f}}+\frac {b (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d^3 \sqrt {f}}+\frac {b (b c-a d) (b e-2 a f) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{2 d^2 f^{3/2}}+\frac {b \left (3 b^2 e^2-8 a b e f+8 a^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{8 d f^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.59, size = 214, normalized size = 0.70 \begin {gather*} \frac {\frac {b^2 d x \sqrt {e+f x^2} \left (12 a d f+b \left (-3 d e-4 c f+2 d f x^2\right )\right )}{f^2}+\frac {8 (b c-a d)^3 \tan ^{-1}\left (\frac {c \sqrt {f}+d x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {c} \sqrt {d e-c f}}\right )}{\sqrt {c} \sqrt {d e-c f}}-\frac {b \left (24 a^2 d^2 f^2-12 a b d f (d e+2 c f)+b^2 \left (3 d^2 e^2+4 c d e f+8 c^2 f^2\right )\right ) \log \left (-\sqrt {f} x+\sqrt {e+f x^2}\right )}{f^{5/2}}}{8 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2*d*x*Sqrt[e + f*x^2]*(12*a*d*f + b*(-3*d*e - 4*c*f + 2*d*f*x^2)))/f^2 + (8*(b*c - a*d)^3*ArcTan[(c*Sqrt[f
] + d*x*(Sqrt[f]*x - Sqrt[e + f*x^2]))/(Sqrt[c]*Sqrt[d*e - c*f])])/(Sqrt[c]*Sqrt[d*e - c*f]) - (b*(24*a^2*d^2*
f^2 - 12*a*b*d*f*(d*e + 2*c*f) + b^2*(3*d^2*e^2 + 4*c*d*e*f + 8*c^2*f^2))*Log[-(Sqrt[f]*x) + Sqrt[e + f*x^2]])
/f^(5/2))/(8*d^3)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(590\) vs. \(2(260)=520\).
time = 0.18, size = 591, normalized size = 1.94

method	result	size
default	\(\frac {b \left (b^{2} d^{2} \left (\frac {x^{3} \sqrt {f \,x^{2}+e}}{4 f}-\frac {3 e \left (\frac {x \sqrt {f \,x^{2}+e}}{2 f}-\frac {e \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{2 f^{\frac {3}{2}}}\right )}{4 f}\right )+\left (3 a b \,d^{2}-b^{2} c d \right ) \left (\frac {x \sqrt {f \,x^{2}+e}}{2 f}-\frac {e \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{2 f^{\frac {3}{2}}}\right )+\frac {3 a^{2} d^{2} \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{\sqrt {f}}-\frac {3 a b c d \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{\sqrt {f}}+\frac {b^{2} c^{2} \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{\sqrt {f}}\right )}{d^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}+\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 d^{3} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}-\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 d^{3} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}\)	\(591\)
risch	\(\text {Expression too large to display}\)	\(1497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/(d*x^2+c)/(f*x^2+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

b/d^3*(b^2*d^2*(1/4*x^3/f*(f*x^2+e)^(1/2)-3/4*e/f*(1/2*x/f*(f*x^2+e)^(1/2)-1/2*e/f^(3/2)*ln(f^(1/2)*x+(f*x^2+e
)^(1/2))))+(3*a*b*d^2-b^2*c*d)*(1/2*x/f*(f*x^2+e)^(1/2)-1/2*e/f^(3/2)*ln(f^(1/2)*x+(f*x^2+e)^(1/2)))+3*a^2*d^2
*ln(f^(1/2)*x+(f*x^2+e)^(1/2))/f^(1/2)-3*a*b*c*d*ln(f^(1/2)*x+(f*x^2+e)^(1/2))/f^(1/2)+b^2*c^2*ln(f^(1/2)*x+(f
*x^2+e)^(1/2))/f^(1/2))-1/2/d^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(1/2
)*ln((-2*(c*f-d*e)/d+2*f*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*f+2*f*
(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-1/2*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*
c^2*d+b^3*c^3)/d^3/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+
2*(-(c*f-d*e)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*f-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(x+(-
c*d)^(1/2)/d))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 10.19, size = 1772, normalized size = 5.83 \begin {gather*} \left [-\frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c^{2} f - c d e} f^{3} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + {\left (3 \, b^{3} c d^{3} e^{3} - 8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2}\right )} f^{3} + 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} f^{2} e + {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f e^{2}\right )} \sqrt {f} \log \left (-2 \, f x^{2} - 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right ) - 2 \, {\left (2 \, b^{3} c^{2} d^{2} f^{3} x^{3} + 3 \, b^{3} c d^{3} f x e^{2} - 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{3} x - {\left (2 \, b^{3} c d^{3} f^{2} x^{3} - {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f^{2} x\right )} e\right )} \sqrt {f x^{2} + e}}{16 \, {\left (c^{2} d^{3} f^{4} - c d^{4} f^{3} e\right )}}, -\frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c^{2} f - c d e} f^{3} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - {\left (3 \, b^{3} c d^{3} e^{3} - 8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2}\right )} f^{3} + 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} f^{2} e + {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f e^{2}\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right ) - {\left (2 \, b^{3} c^{2} d^{2} f^{3} x^{3} + 3 \, b^{3} c d^{3} f x e^{2} - 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{3} x - {\left (2 \, b^{3} c d^{3} f^{2} x^{3} - {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f^{2} x\right )} e\right )} \sqrt {f x^{2} + e}}{8 \, {\left (c^{2} d^{3} f^{4} - c d^{4} f^{3} e\right )}}, \frac {8 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c^{2} f + c d e} f^{3} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) - {\left (3 \, b^{3} c d^{3} e^{3} - 8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2}\right )} f^{3} + 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} f^{2} e + {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f e^{2}\right )} \sqrt {f} \log \left (-2 \, f x^{2} - 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right ) + 2 \, {\left (2 \, b^{3} c^{2} d^{2} f^{3} x^{3} + 3 \, b^{3} c d^{3} f x e^{2} - 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{3} x - {\left (2 \, b^{3} c d^{3} f^{2} x^{3} - {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f^{2} x\right )} e\right )} \sqrt {f x^{2} + e}}{16 \, {\left (c^{2} d^{3} f^{4} - c d^{4} f^{3} e\right )}}, \frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c^{2} f + c d e} f^{3} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) + {\left (3 \, b^{3} c d^{3} e^{3} - 8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2}\right )} f^{3} + 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} f^{2} e + {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f e^{2}\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right ) + {\left (2 \, b^{3} c^{2} d^{2} f^{3} x^{3} + 3 \, b^{3} c d^{3} f x e^{2} - 4 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{3} x - {\left (2 \, b^{3} c d^{3} f^{2} x^{3} - {\left (b^{3} c^{2} d^{2} - 12 \, a b^{2} c d^{3}\right )} f^{2} x\right )} e\right )} \sqrt {f x^{2} + e}}{8 \, {\left (c^{2} d^{3} f^{4} - c d^{4} f^{3} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c^2*f - c*d*e)*f^3*log((8*c^2*f^2*x^4 + 4*(
2*c*f*x^3 - (d*x^3 - c*x)*e)*sqrt(c^2*f - c*d*e)*sqrt(f*x^2 + e) + (d^2*x^4 - 6*c*d*x^2 + c^2)*e^2 - 8*(c*d*f*
x^4 - c^2*f*x^2)*e)/(d^2*x^4 + 2*c*d*x^2 + c^2)) + (3*b^3*c*d^3*e^3 - 8*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2
*d^2)*f^3 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*f^2*e + (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f*e^2)*sqrt
(f)*log(-2*f*x^2 - 2*sqrt(f*x^2 + e)*sqrt(f)*x - e) - 2*(2*b^3*c^2*d^2*f^3*x^3 + 3*b^3*c*d^3*f*x*e^2 - 4*(b^3*
c^3*d - 3*a*b^2*c^2*d^2)*f^3*x - (2*b^3*c*d^3*f^2*x^3 - (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f^2*x)*e)*sqrt(f*x^2 +
e))/(c^2*d^3*f^4 - c*d^4*f^3*e), -1/8*(2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c^2*f - c*d*
e)*f^3*log((8*c^2*f^2*x^4 + 4*(2*c*f*x^3 - (d*x^3 - c*x)*e)*sqrt(c^2*f - c*d*e)*sqrt(f*x^2 + e) + (d^2*x^4 - 6
*c*d*x^2 + c^2)*e^2 - 8*(c*d*f*x^4 - c^2*f*x^2)*e)/(d^2*x^4 + 2*c*d*x^2 + c^2)) - (3*b^3*c*d^3*e^3 - 8*(b^3*c^
4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2)*f^3 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*f^2*e + (b^3*c^2*d^
2 - 12*a*b^2*c*d^3)*f*e^2)*sqrt(-f)*arctan(sqrt(-f)*x/sqrt(f*x^2 + e)) - (2*b^3*c^2*d^2*f^3*x^3 + 3*b^3*c*d^3*
f*x*e^2 - 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2)*f^3*x - (2*b^3*c*d^3*f^2*x^3 - (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f^2*x)
*e)*sqrt(f*x^2 + e))/(c^2*d^3*f^4 - c*d^4*f^3*e), 1/16*(8*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
sqrt(-c^2*f + c*d*e)*f^3*arctan(1/2*(2*c*f*x^2 - (d*x^2 - c)*e)*sqrt(-c^2*f + c*d*e)*sqrt(f*x^2 + e)/(c^2*f^2*
x^3 - c*d*x*e^2 - (c*d*f*x^3 - c^2*f*x)*e)) - (3*b^3*c*d^3*e^3 - 8*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2)
*f^3 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*f^2*e + (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f*e^2)*sqrt(f)*l
og(-2*f*x^2 - 2*sqrt(f*x^2 + e)*sqrt(f)*x - e) + 2*(2*b^3*c^2*d^2*f^3*x^3 + 3*b^3*c*d^3*f*x*e^2 - 4*(b^3*c^3*d
- 3*a*b^2*c^2*d^2)*f^3*x - (2*b^3*c*d^3*f^2*x^3 - (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f^2*x)*e)*sqrt(f*x^2 + e))/(
c^2*d^3*f^4 - c*d^4*f^3*e), 1/8*(4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-c^2*f + c*d*e)*f^
3*arctan(1/2*(2*c*f*x^2 - (d*x^2 - c)*e)*sqrt(-c^2*f + c*d*e)*sqrt(f*x^2 + e)/(c^2*f^2*x^3 - c*d*x*e^2 - (c*d*
f*x^3 - c^2*f*x)*e)) + (3*b^3*c*d^3*e^3 - 8*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2)*f^3 + 4*(b^3*c^3*d - 3
*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*f^2*e + (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f*e^2)*sqrt(-f)*arctan(sqrt(-f)*x/sqrt(
f*x^2 + e)) + (2*b^3*c^2*d^2*f^3*x^3 + 3*b^3*c*d^3*f*x*e^2 - 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2)*f^3*x - (2*b^3*c*
d^3*f^2*x^3 - (b^3*c^2*d^2 - 12*a*b^2*c*d^3)*f^2*x)*e)*sqrt(f*x^2 + e))/(c^2*d^3*f^4 - c*d^4*f^3*e)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{3}}{\left (c + d x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^3}{\left (d\,x^2+c\right )\,\sqrt {f\,x^2+e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________