3.8.34 \(\int x^5 (a+b x^2+c x^4)^{3/2} \, dx\)

Optimal. Leaf size=204 \[ \frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2048 c^{9/2}}-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c} \]

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Rubi [A]  time = 0.18, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1114, 742, 640, 612, 621, 206} \begin {gather*} \frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 c^4}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2048 c^{9/2}}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

-((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(1024*c^4) + ((7*b^2 - 4*a*c)*(b + 2*c*
x^2)*(a + b*x^2 + c*x^4)^(3/2))/(384*c^3) - (7*b*(a + b*x^2 + c*x^4)^(5/2))/(120*c^2) + (x^2*(a + b*x^2 + c*x^
4)^(5/2))/(12*c) + ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]
)/(2048*c^(9/2))

Rule 206

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 640

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

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac {\operatorname {Subst}\left (\int \left (-a-\frac {7 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{12 c}\\ &=-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac {\left (7 b^2-4 a c\right ) \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{48 c^2}\\ &=\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{256 c^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2048 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{1024 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2048 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 175, normalized size = 0.86 \begin {gather*} \frac {\frac {\left (7 b^2-4 a c\right ) \left (2 \sqrt {c} \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4} \left (4 c \left (5 a+2 c x^4\right )-3 b^2+8 b c x^2\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\right )}{512 c^{7/2}}+x^2 \left (a+b x^2+c x^4\right )^{5/2}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{10 c}}{12 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

((-7*b*(a + b*x^2 + c*x^4)^(5/2))/(10*c) + x^2*(a + b*x^2 + c*x^4)^(5/2) + ((7*b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*
c*x^2)*Sqrt[a + b*x^2 + c*x^4]*(-3*b^2 + 8*b*c*x^2 + 4*c*(5*a + 2*c*x^4)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c
*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]))/(512*c^(7/2)))/(12*c)

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IntegrateAlgebraic [A]  time = 0.77, size = 209, normalized size = 1.02 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-1296 a^2 b c^2+480 a^2 c^3 x^2+760 a b^3 c-432 a b^2 c^2 x^2+288 a b c^3 x^4+2240 a c^4 x^6-105 b^5+70 b^4 c x^2-56 b^3 c^2 x^4+48 b^2 c^3 x^6+1664 b c^4 x^8+1280 c^5 x^{10}\right )}{15360 c^4}+\frac {\left (64 a^3 c^3-144 a^2 b^2 c^2+60 a b^4 c-7 b^6\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x^2+c x^4}+b+2 c x^2\right )}{2048 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(Sqrt[a + b*x^2 + c*x^4]*(-105*b^5 + 760*a*b^3*c - 1296*a^2*b*c^2 + 70*b^4*c*x^2 - 432*a*b^2*c^2*x^2 + 480*a^2
*c^3*x^2 - 56*b^3*c^2*x^4 + 288*a*b*c^3*x^4 + 48*b^2*c^3*x^6 + 2240*a*c^4*x^6 + 1664*b*c^4*x^8 + 1280*c^5*x^10
))/(15360*c^4) + ((-7*b^6 + 60*a*b^4*c - 144*a^2*b^2*c^2 + 64*a^3*c^3)*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*
x^2 + c*x^4]])/(2048*c^(9/2))

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fricas [A]  time = 2.37, size = 451, normalized size = 2.21 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (1280 \, c^{6} x^{10} + 1664 \, b c^{5} x^{8} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{61440 \, c^{5}}, -\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \, {\left (1280 \, c^{6} x^{10} + 1664 \, b c^{5} x^{8} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30720 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/61440*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c)*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 + 4
*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) - 4*(1280*c^6*x^10 + 1664*b*c^5*x^8 + 16*(3*b^2*c^4 +
140*a*c^5)*x^6 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^4 + 2*(35*b^4*c^2 -
216*a*b^2*c^3 + 240*a^2*c^4)*x^2)*sqrt(c*x^4 + b*x^2 + a))/c^5, -1/30720*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^
2*c^2 - 64*a^3*c^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 + b*c*x^2 + a*
c)) - 2*(1280*c^6*x^10 + 1664*b*c^5*x^8 + 16*(3*b^2*c^4 + 140*a*c^5)*x^6 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^
2*b*c^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^4 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x^2)*sqrt(c*x^4 + b*x^
2 + a))/c^5]

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giac [B]  time = 0.40, size = 535, normalized size = 2.62 \begin {gather*} \frac {1}{768} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {5 \, b^{2} c - 12 \, a c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, b^{3} - 52 \, a b c}{c^{3}}\right )} + \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {7}{2}}}\right )} a + \frac {1}{7680} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {9}{2}}}\right )} b + \frac {1}{30720} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {9 \, b^{2} c^{3} - 20 \, a c^{4}}{c^{5}}\right )} x^{2} + \frac {21 \, b^{3} c^{2} - 68 \, a b c^{3}}{c^{5}}\right )} x^{2} - \frac {105 \, b^{4} c - 448 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}}{c^{5}}\right )} x^{2} + \frac {315 \, b^{5} - 1680 \, a b^{3} c + 1808 \, a^{2} b c^{2}}{c^{5}}\right )} + \frac {15 \, {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {11}{2}}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

1/768*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(6*x^2 + b/c)*x^2 - (5*b^2*c - 12*a*c^2)/c^3)*x^2 + (15*b^3 - 52*a*b*c)
/c^3) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*log(abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/c
^(7/2))*a + 1/7680*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(6*(8*x^2 + b/c)*x^2 - (7*b^2*c^2 - 16*a*c^3)/c^4)*x^2 + (
35*b^3*c - 116*a*b*c^2)/c^4)*x^2 - (105*b^4 - 460*a*b^2*c + 256*a^2*c^2)/c^4) - 15*(7*b^5 - 40*a*b^3*c + 48*a^
2*b*c^2)*log(abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/c^(9/2))*b + 1/30720*(2*sqrt(c*x^4 +
b*x^2 + a)*(2*(4*(2*(8*(10*x^2 + b/c)*x^2 - (9*b^2*c^3 - 20*a*c^4)/c^5)*x^2 + (21*b^3*c^2 - 68*a*b*c^3)/c^5)*
x^2 - (105*b^4*c - 448*a*b^2*c^2 + 240*a^2*c^3)/c^5)*x^2 + (315*b^5 - 1680*a*b^3*c + 1808*a^2*b*c^2)/c^5) + 15
*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*log(abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt
(c) - b))/c^(11/2))*c

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maple [B]  time = 0.02, size = 432, normalized size = 2.12 \begin {gather*} \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{10}}{12}+\frac {13 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{8}}{120}+\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,x^{6}}{48}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} x^{6}}{320 c}+\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,x^{4}}{160 c}-\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} x^{4}}{1920 c^{2}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} x^{2}}{32 c}-\frac {9 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{2} x^{2}}{320 c^{2}}+\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4} x^{2}}{1536 c^{3}}-\frac {a^{3} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {9 a^{2} b^{2} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{128 c^{\frac {5}{2}}}-\frac {15 a \,b^{4} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}+\frac {7 b^{6} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2048 c^{\frac {9}{2}}}-\frac {27 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} b}{320 c^{2}}+\frac {19 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{3}}{384 c^{3}}-\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{5}}{1024 c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(c*x^4+b*x^2+a)^(3/2),x)

[Out]

7/1536*b^4/c^3*x^2*(c*x^4+b*x^2+a)^(1/2)+1/320*b^2*x^6/c*(c*x^4+b*x^2+a)^(1/2)-7/1920*b^3/c^2*x^4*(c*x^4+b*x^2
+a)^(1/2)+1/32*a^2*x^2/c*(c*x^4+b*x^2+a)^(1/2)-15/512*a*b^4/c^(7/2)*ln((c*x^2+1/2*b)/c^(1/2)+(c*x^4+b*x^2+a)^(
1/2))+9/128*a^2*b^2/c^(5/2)*ln((c*x^2+1/2*b)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-27/320*a^2*b/c^2*(c*x^4+b*x^2+a)^(
1/2)+19/384*a*b^3/c^3*(c*x^4+b*x^2+a)^(1/2)+13/120*b*x^8*(c*x^4+b*x^2+a)^(1/2)+3/160*a*b*x^4/c*(c*x^4+b*x^2+a)
^(1/2)-9/320*a*b^2/c^2*x^2*(c*x^4+b*x^2+a)^(1/2)-1/32*a^3/c^(3/2)*ln((c*x^2+1/2*b)/c^(1/2)+(c*x^4+b*x^2+a)^(1/
2))+1/12*c*x^10*(c*x^4+b*x^2+a)^(1/2)+7/48*a*x^6*(c*x^4+b*x^2+a)^(1/2)-7/1024*b^5/c^4*(c*x^4+b*x^2+a)^(1/2)+7/
2048*b^6/c^(9/2)*ln((c*x^2+1/2*b)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)^(3/2),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(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^2 + c*x^4)^(3/2),x)

[Out]

int(x^5*(a + b*x^2 + c*x^4)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x**5*(a + b*x**2 + c*x**4)**(3/2), x)

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