\(\int (d+e x^2)^{3/2} (a+b x^2+c x^4) \, dx\) [277]

Optimal result

Integrand size = 24, antiderivative size = 175 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {d \left (3 c d^2-8 b d e+48 a e^2\right ) x \sqrt {d+e x^2}}{128 e^2}+\frac {\left (3 c d^2-8 b d e+48 a e^2\right ) x \left (d+e x^2\right )^{3/2}}{192 e^2}-\frac {(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac {d^2 \left (3 c d^2-8 b d e+48 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{128 e^{5/2}} \]

[Out]

1/192*(48*a*e^2-8*b*d*e+3*c*d^2)*x*(e*x^2+d)^(3/2)/e^2-1/48*(-8*b*e+3*c*d)*x*(e*x^2+d)^(5/2)/e^2+1/8*c*x^3*(e*
x^2+d)^(5/2)/e+1/128*d^2*(48*a*e^2-8*b*d*e+3*c*d^2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(5/2)+1/128*d*(48*a*e
^2-8*b*d*e+3*c*d^2)*x*(e*x^2+d)^(1/2)/e^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1173, 396, 201, 223, 212} \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {d^2 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}+\frac {x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac {d x \sqrt {d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}-\frac {x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]

[In]

Int[(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x]

[Out]

(d*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*Sqrt[d + e*x^2])/(128*e^2) + ((3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*(d + e*x^2)
^(3/2))/(192*e^2) - ((3*c*d - 8*b*e)*x*(d + e*x^2)^(5/2))/(48*e^2) + (c*x^3*(d + e*x^2)^(5/2))/(8*e) + (d^2*(3
*c*d^2 - 8*b*d*e + 48*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(5/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 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 1173

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*x^(4*p - 1)*(
(d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*
p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /
; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[
q, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac {\int \left (d+e x^2\right )^{3/2} \left (8 a e-(3 c d-8 b e) x^2\right ) \, dx}{8 e} \\ & = -\frac {(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}-\frac {1}{48} \left (-48 a-\frac {d (3 c d-8 b e)}{e^2}\right ) \int \left (d+e x^2\right )^{3/2} \, dx \\ & = \frac {1}{192} \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac {(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac {1}{64} \left (d \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right )\right ) \int \sqrt {d+e x^2} \, dx \\ & = \frac {1}{128} d \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \sqrt {d+e x^2}+\frac {1}{192} \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac {(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac {1}{128} \left (d^2 \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx \\ & = \frac {1}{128} d \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \sqrt {d+e x^2}+\frac {1}{192} \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac {(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac {1}{128} \left (d^2 \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right ) \\ & = \frac {1}{128} d \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \sqrt {d+e x^2}+\frac {1}{192} \left (48 a+\frac {d (3 c d-8 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}-\frac {(3 c d-8 b e) x \left (d+e x^2\right )^{5/2}}{48 e^2}+\frac {c x^3 \left (d+e x^2\right )^{5/2}}{8 e}+\frac {d^2 \left (3 c d^2-8 b d e+48 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{128 e^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (c \left (-9 d^3+6 d^2 e x^2+72 d e^2 x^4+48 e^3 x^6\right )+8 e \left (6 a e \left (5 d+2 e x^2\right )+b \left (3 d^2+14 d e x^2+8 e^2 x^4\right )\right )\right )-3 \left (3 c d^4-8 d^2 e (b d-6 a e)\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{384 e^{5/2}} \]

[In]

Integrate[(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x]

[Out]

(Sqrt[e]*x*Sqrt[d + e*x^2]*(c*(-9*d^3 + 6*d^2*e*x^2 + 72*d*e^2*x^4 + 48*e^3*x^6) + 8*e*(6*a*e*(5*d + 2*e*x^2)
+ b*(3*d^2 + 14*d*e*x^2 + 8*e^2*x^4))) - 3*(3*c*d^4 - 8*d^2*e*(b*d - 6*a*e))*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2
]])/(384*e^(5/2))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(\frac {\frac {3 \left (a \,e^{2}-\frac {1}{6} b d e +\frac {1}{16} c \,d^{2}\right ) d^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )}{8}+\frac {5 \left (d \left (\frac {3}{10} c \,x^{4}+\frac {7}{15} b \,x^{2}+a \right ) e^{\frac {5}{2}}+\frac {2 x^{2} \left (\frac {1}{2} c \,x^{4}+\frac {2}{3} b \,x^{2}+a \right ) e^{\frac {7}{2}}}{5}+\frac {\left (\left (\frac {c \,x^{2}}{4}+b \right ) e^{\frac {3}{2}}-\frac {3 c d \sqrt {e}}{8}\right ) d^{2}}{10}\right ) \sqrt {e \,x^{2}+d}\, x}{8}}{e^{\frac {5}{2}}}\)	\(124\)
risch	\(\frac {x \left (48 e^{3} c \,x^{6}+64 e^{3} b \,x^{4}+72 d \,e^{2} c \,x^{4}+96 a \,e^{3} x^{2}+112 d \,e^{2} b \,x^{2}+6 d^{2} e \,x^{2} c +240 d \,e^{2} a +24 d^{2} e b -9 d^{3} c \right ) \sqrt {e \,x^{2}+d}}{384 e^{2}}+\frac {d^{2} \left (48 a \,e^{2}-8 b d e +3 c \,d^{2}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{128 e^{\frac {5}{2}}}\)	\(137\)
default	\(a \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )+c \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{8 e}-\frac {3 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6 e}\right )}{8 e}\right )+b \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6 e}\right )\)	\(229\)

[In]

int((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

5/8*(3/5*(a*e^2-1/6*b*d*e+1/16*c*d^2)*d^2*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+(d*(3/10*c*x^4+7/15*b*x^2+a)*e^(5
/2)+2/5*x^2*(1/2*c*x^4+2/3*b*x^2+a)*e^(7/2)+1/10*((1/4*c*x^2+b)*e^(3/2)-3/8*c*d*e^(1/2))*d^2)*(e*x^2+d)^(1/2)*
x)/e^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.74 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\left [\frac {3 \, {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (48 \, c e^{4} x^{7} + 8 \, {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} x^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{768 \, e^{3}}, -\frac {3 \, {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (48 \, c e^{4} x^{7} + 8 \, {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} x^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{384 \, e^{3}}\right ] \]

[In]

integrate((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[1/768*(3*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(48
*c*e^4*x^7 + 8*(9*c*d*e^3 + 8*b*e^4)*x^5 + 2*(3*c*d^2*e^2 + 56*b*d*e^3 + 48*a*e^4)*x^3 - 3*(3*c*d^3*e - 8*b*d^
2*e^2 - 80*a*d*e^3)*x)*sqrt(e*x^2 + d))/e^3, -1/384*(3*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*sqrt(-e)*arctan(sq
rt(-e)*x/sqrt(e*x^2 + d)) - (48*c*e^4*x^7 + 8*(9*c*d*e^3 + 8*b*e^4)*x^5 + 2*(3*c*d^2*e^2 + 56*b*d*e^3 + 48*a*e
^4)*x^3 - 3*(3*c*d^3*e - 8*b*d^2*e^2 - 80*a*d*e^3)*x)*sqrt(e*x^2 + d))/e^3]

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.57 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\begin {cases} \sqrt {d + e x^{2}} \left (\frac {c e x^{7}}{8} + \frac {x^{5} \left (b e^{2} + \frac {9 c d e}{8}\right )}{6 e} + \frac {x^{3} \left (a e^{2} + 2 b d e + c d^{2} - \frac {5 d \left (b e^{2} + \frac {9 c d e}{8}\right )}{6 e}\right )}{4 e} + \frac {x \left (2 a d e + b d^{2} - \frac {3 d \left (a e^{2} + 2 b d e + c d^{2} - \frac {5 d \left (b e^{2} + \frac {9 c d e}{8}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) + \left (a d^{2} - \frac {d \left (2 a d e + b d^{2} - \frac {3 d \left (a e^{2} + 2 b d e + c d^{2} - \frac {5 d \left (b e^{2} + \frac {9 c d e}{8}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {e} \sqrt {d + e x^{2}} + 2 e x \right )}}{\sqrt {e}} & \text {for}\: d \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {e x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a x + \frac {b x^{3}}{3} + \frac {c x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x**2+d)**(3/2)*(c*x**4+b*x**2+a),x)

[Out]

Piecewise((sqrt(d + e*x**2)*(c*e*x**7/8 + x**5*(b*e**2 + 9*c*d*e/8)/(6*e) + x**3*(a*e**2 + 2*b*d*e + c*d**2 -
5*d*(b*e**2 + 9*c*d*e/8)/(6*e))/(4*e) + x*(2*a*d*e + b*d**2 - 3*d*(a*e**2 + 2*b*d*e + c*d**2 - 5*d*(b*e**2 + 9
*c*d*e/8)/(6*e))/(4*e))/(2*e)) + (a*d**2 - d*(2*a*d*e + b*d**2 - 3*d*(a*e**2 + 2*b*d*e + c*d**2 - 5*d*(b*e**2
+ 9*c*d*e/8)/(6*e))/(4*e))/(2*e))*Piecewise((log(2*sqrt(e)*sqrt(d + e*x**2) + 2*e*x)/sqrt(e), Ne(d, 0)), (x*lo
g(x)/sqrt(e*x**2), True)), Ne(e, 0)), (d**(3/2)*(a*x + b*x**3/3 + c*x**5/5), True))

Maxima [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.89 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, c e x^{2} + \frac {9 \, c d e^{6} + 8 \, b e^{7}}{e^{6}}\right )} x^{2} + \frac {3 \, c d^{2} e^{5} + 56 \, b d e^{6} + 48 \, a e^{7}}{e^{6}}\right )} x^{2} - \frac {3 \, {\left (3 \, c d^{3} e^{4} - 8 \, b d^{2} e^{5} - 80 \, a d e^{6}\right )}}{e^{6}}\right )} \sqrt {e x^{2} + d} x - \frac {{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{128 \, e^{\frac {5}{2}}} \]

[In]

integrate((e*x^2+d)^(3/2)*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*c*e*x^2 + (9*c*d*e^6 + 8*b*e^7)/e^6)*x^2 + (3*c*d^2*e^5 + 56*b*d*e^6 + 48*a*e^7)/e^6)*x^2 - 3*(
3*c*d^3*e^4 - 8*b*d^2*e^5 - 80*a*d*e^6)/e^6)*sqrt(e*x^2 + d)*x - 1/128*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*lo
g(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/e^(5/2)

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right ) \, dx=\int {\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right ) \,d x \]

[In]

int((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4),x)

[Out]

int((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4), x)