3.2.99 \(\int (d+e x^2)^{5/2} (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=215 \[ \frac {x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac {d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac {d^2 x \sqrt {d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac {d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}-\frac {x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \]

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Rubi [A]  time = 0.16, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1159, 388, 195, 217, 206} \begin {gather*} \frac {x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac {d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac {d^2 x \sqrt {d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac {d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}-\frac {x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^(5/2)*(a + b*x^2 + c*x^4),x]

[Out]

(d^2*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*Sqrt[d + e*x^2])/(256*e^2) + (d*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d +
e*x^2)^(3/2))/(384*e^2) + ((3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(5/2))/(480*e^2) - ((3*c*d - 10*b*e)*
x*(d + e*x^2)^(7/2))/(80*e^2) + (c*x^3*(d + e*x^2)^(7/2))/(10*e) + (d^3*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*ArcTan
h[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(256*e^(5/2))

Rule 195

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 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 217

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 388

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 1159

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*} \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx &=\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {\int \left (d+e x^2\right )^{5/2} \left (10 a e-(3 c d-10 b e) x^2\right ) \, dx}{10 e}\\ &=-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}-\frac {1}{80} \left (-80 a-\frac {d (3 c d-10 b e)}{e^2}\right ) \int \left (d+e x^2\right )^{5/2} \, dx\\ &=\frac {1}{480} \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {1}{96} \left (d \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right )\right ) \int \left (d+e x^2\right )^{3/2} \, dx\\ &=\frac {1}{384} d \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac {1}{480} \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {1}{128} \left (d^2 \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right )\right ) \int \sqrt {d+e x^2} \, dx\\ &=\frac {1}{256} d^2 \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \sqrt {d+e x^2}+\frac {1}{384} d \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac {1}{480} \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {1}{256} \left (d^3 \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx\\ &=\frac {1}{256} d^2 \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \sqrt {d+e x^2}+\frac {1}{384} d \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac {1}{480} \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {1}{256} \left (d^3 \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )\\ &=\frac {1}{256} d^2 \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \sqrt {d+e x^2}+\frac {1}{384} d \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac {1}{480} \left (80 a+\frac {d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {d^3 \left (3 c d^2-10 b d e+80 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{256 e^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 190, normalized size = 0.88 \begin {gather*} \frac {\sqrt {d+e x^2} \left (\frac {15 d^{5/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (10 e (8 a e-b d)+3 c d^2\right )}{\sqrt {\frac {e x^2}{d}+1}}+\sqrt {e} x \left (10 e \left (8 a e \left (33 d^2+26 d e x^2+8 e^2 x^4\right )+b \left (15 d^3+118 d^2 e x^2+136 d e^2 x^4+48 e^3 x^6\right )\right )+c \left (-45 d^4+30 d^3 e x^2+744 d^2 e^2 x^4+1008 d e^3 x^6+384 e^4 x^8\right )\right )\right )}{3840 e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(Sqrt[e]*x*(c*(-45*d^4 + 30*d^3*e*x^2 + 744*d^2*e^2*x^4 + 1008*d*e^3*x^6 + 384*e^4*x^8) + 10*
e*(8*a*e*(33*d^2 + 26*d*e*x^2 + 8*e^2*x^4) + b*(15*d^3 + 118*d^2*e*x^2 + 136*d*e^2*x^4 + 48*e^3*x^6))) + (15*d
^(5/2)*(3*c*d^2 + 10*e*(-(b*d) + 8*a*e))*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[1 + (e*x^2)/d]))/(3840*e^(5/2))

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IntegrateAlgebraic [A]  time = 0.41, size = 189, normalized size = 0.88 \begin {gather*} \frac {\log \left (\sqrt {d+e x^2}-\sqrt {e} x\right ) \left (-80 a d^3 e^2+10 b d^4 e-3 c d^5\right )}{256 e^{5/2}}+\frac {\sqrt {d+e x^2} \left (2640 a d^2 e^2 x+2080 a d e^3 x^3+640 a e^4 x^5+150 b d^3 e x+1180 b d^2 e^2 x^3+1360 b d e^3 x^5+480 b e^4 x^7-45 c d^4 x+30 c d^3 e x^3+744 c d^2 e^2 x^5+1008 c d e^3 x^7+384 c e^4 x^9\right )}{3840 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(-45*c*d^4*x + 150*b*d^3*e*x + 2640*a*d^2*e^2*x + 30*c*d^3*e*x^3 + 1180*b*d^2*e^2*x^3 + 2080*
a*d*e^3*x^3 + 744*c*d^2*e^2*x^5 + 1360*b*d*e^3*x^5 + 640*a*e^4*x^5 + 1008*c*d*e^3*x^7 + 480*b*e^4*x^7 + 384*c*
e^4*x^9))/(3840*e^2) + ((-3*c*d^5 + 10*b*d^4*e - 80*a*d^3*e^2)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/(256*e^(5/
2))

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fricas [A]  time = 1.65, size = 370, normalized size = 1.72 \begin {gather*} \left [\frac {15 \, {\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (384 \, c e^{5} x^{9} + 48 \, {\left (21 \, c d e^{4} + 10 \, b e^{5}\right )} x^{7} + 8 \, {\left (93 \, c d^{2} e^{3} + 170 \, b d e^{4} + 80 \, a e^{5}\right )} x^{5} + 10 \, {\left (3 \, c d^{3} e^{2} + 118 \, b d^{2} e^{3} + 208 \, a d e^{4}\right )} x^{3} - 15 \, {\left (3 \, c d^{4} e - 10 \, b d^{3} e^{2} - 176 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{7680 \, e^{3}}, -\frac {15 \, {\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (384 \, c e^{5} x^{9} + 48 \, {\left (21 \, c d e^{4} + 10 \, b e^{5}\right )} x^{7} + 8 \, {\left (93 \, c d^{2} e^{3} + 170 \, b d e^{4} + 80 \, a e^{5}\right )} x^{5} + 10 \, {\left (3 \, c d^{3} e^{2} + 118 \, b d^{2} e^{3} + 208 \, a d e^{4}\right )} x^{3} - 15 \, {\left (3 \, c d^{4} e - 10 \, b d^{3} e^{2} - 176 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{3840 \, e^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/7680*(15*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*
(384*c*e^5*x^9 + 48*(21*c*d*e^4 + 10*b*e^5)*x^7 + 8*(93*c*d^2*e^3 + 170*b*d*e^4 + 80*a*e^5)*x^5 + 10*(3*c*d^3*
e^2 + 118*b*d^2*e^3 + 208*a*d*e^4)*x^3 - 15*(3*c*d^4*e - 10*b*d^3*e^2 - 176*a*d^2*e^3)*x)*sqrt(e*x^2 + d))/e^3
, -1/3840*(15*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (384*c*e^5*x
^9 + 48*(21*c*d*e^4 + 10*b*e^5)*x^7 + 8*(93*c*d^2*e^3 + 170*b*d*e^4 + 80*a*e^5)*x^5 + 10*(3*c*d^3*e^2 + 118*b*
d^2*e^3 + 208*a*d*e^4)*x^3 - 15*(3*c*d^4*e - 10*b*d^3*e^2 - 176*a*d^2*e^3)*x)*sqrt(e*x^2 + d))/e^3]

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giac [A]  time = 0.23, size = 180, normalized size = 0.84 \begin {gather*} -\frac {1}{256} \, {\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -x e^{\frac {1}{2}} + \sqrt {x^{2} e + d} \right |}\right ) + \frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c x^{2} e^{2} + {\left (21 \, c d e^{9} + 10 \, b e^{10}\right )} e^{\left (-8\right )}\right )} x^{2} + {\left (93 \, c d^{2} e^{8} + 170 \, b d e^{9} + 80 \, a e^{10}\right )} e^{\left (-8\right )}\right )} x^{2} + 5 \, {\left (3 \, c d^{3} e^{7} + 118 \, b d^{2} e^{8} + 208 \, a d e^{9}\right )} e^{\left (-8\right )}\right )} x^{2} - 15 \, {\left (3 \, c d^{4} e^{6} - 10 \, b d^{3} e^{7} - 176 \, a d^{2} e^{8}\right )} e^{\left (-8\right )}\right )} \sqrt {x^{2} e + d} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*e^(-5/2)*log(abs(-x*e^(1/2) + sqrt(x^2*e + d))) + 1/3840*(2*(4*(6
*(8*c*x^2*e^2 + (21*c*d*e^9 + 10*b*e^10)*e^(-8))*x^2 + (93*c*d^2*e^8 + 170*b*d*e^9 + 80*a*e^10)*e^(-8))*x^2 +
5*(3*c*d^3*e^7 + 118*b*d^2*e^8 + 208*a*d*e^9)*e^(-8))*x^2 - 15*(3*c*d^4*e^6 - 10*b*d^3*e^7 - 176*a*d^2*e^8)*e^
(-8))*sqrt(x^2*e + d)*x

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maple [A]  time = 0.01, size = 283, normalized size = 1.32 \begin {gather*} \frac {5 a \,d^{3} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{16 \sqrt {e}}-\frac {5 b \,d^{4} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{128 e^{\frac {3}{2}}}+\frac {3 c \,d^{5} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{256 e^{\frac {5}{2}}}+\frac {5 \sqrt {e \,x^{2}+d}\, a \,d^{2} x}{16}-\frac {5 \sqrt {e \,x^{2}+d}\, b \,d^{3} x}{128 e}+\frac {3 \sqrt {e \,x^{2}+d}\, c \,d^{4} x}{256 e^{2}}+\frac {5 \left (e \,x^{2}+d \right )^{\frac {3}{2}} a d x}{24}-\frac {5 \left (e \,x^{2}+d \right )^{\frac {3}{2}} b \,d^{2} x}{192 e}+\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} c \,d^{3} x}{128 e^{2}}+\frac {\left (e \,x^{2}+d \right )^{\frac {7}{2}} c \,x^{3}}{10 e}+\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} a x}{6}-\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} b d x}{48 e}+\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}} c \,d^{2} x}{160 e^{2}}+\frac {\left (e \,x^{2}+d \right )^{\frac {7}{2}} b x}{8 e}-\frac {3 \left (e \,x^{2}+d \right )^{\frac {7}{2}} c d x}{80 e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c*x^3*(e*x^2+d)^(7/2)/e-3/80*c*d/e^2*x*(e*x^2+d)^(7/2)+1/160*c*d^2/e^2*x*(e*x^2+d)^(5/2)+1/128*c*d^3/e^2*
x*(e*x^2+d)^(3/2)+3/256*c*d^4/e^2*x*(e*x^2+d)^(1/2)+3/256*c*d^5/e^(5/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/8*b*x*
(e*x^2+d)^(7/2)/e-1/48*b*d/e*x*(e*x^2+d)^(5/2)-5/192*b*d^2/e*x*(e*x^2+d)^(3/2)-5/128*b*d^3/e*x*(e*x^2+d)^(1/2)
-5/128*b*d^4/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/6*a*x*(e*x^2+d)^(5/2)+5/24*a*d*x*(e*x^2+d)^(3/2)+5/16*a*d
^2*x*(e*x^2+d)^(1/2)+5/16*a*d^3/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))

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maxima [A]  time = 1.12, size = 261, normalized size = 1.21 \begin {gather*} \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}} c x^{3}}{10 \, e} + \frac {1}{6} \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} a x + \frac {5}{24} \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} a d x + \frac {5}{16} \, \sqrt {e x^{2} + d} a d^{2} x - \frac {3 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} c d x}{80 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} c d^{2} x}{160 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} c d^{3} x}{128 \, e^{2}} + \frac {3 \, \sqrt {e x^{2} + d} c d^{4} x}{256 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}} b x}{8 \, e} - \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} b d x}{48 \, e} - \frac {5 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} b d^{2} x}{192 \, e} - \frac {5 \, \sqrt {e x^{2} + d} b d^{3} x}{128 \, e} + \frac {3 \, c d^{5} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{256 \, e^{\frac {5}{2}}} - \frac {5 \, b d^{4} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{128 \, e^{\frac {3}{2}}} + \frac {5 \, a d^{3} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{16 \, \sqrt {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

1/10*(e*x^2 + d)^(7/2)*c*x^3/e + 1/6*(e*x^2 + d)^(5/2)*a*x + 5/24*(e*x^2 + d)^(3/2)*a*d*x + 5/16*sqrt(e*x^2 +
d)*a*d^2*x - 3/80*(e*x^2 + d)^(7/2)*c*d*x/e^2 + 1/160*(e*x^2 + d)^(5/2)*c*d^2*x/e^2 + 1/128*(e*x^2 + d)^(3/2)*
c*d^3*x/e^2 + 3/256*sqrt(e*x^2 + d)*c*d^4*x/e^2 + 1/8*(e*x^2 + d)^(7/2)*b*x/e - 1/48*(e*x^2 + d)^(5/2)*b*d*x/e
- 5/192*(e*x^2 + d)^(3/2)*b*d^2*x/e - 5/128*sqrt(e*x^2 + d)*b*d^3*x/e + 3/256*c*d^5*arcsinh(e*x/sqrt(d*e))/e^
(5/2) - 5/128*b*d^4*arcsinh(e*x/sqrt(d*e))/e^(3/2) + 5/16*a*d^3*arcsinh(e*x/sqrt(d*e))/sqrt(e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 63.83, size = 505, normalized size = 2.35 \begin {gather*} \frac {a d^{\frac {5}{2}} x \sqrt {1 + \frac {e x^{2}}{d}}}{2} + \frac {3 a d^{\frac {5}{2}} x}{16 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {35 a d^{\frac {3}{2}} e x^{3}}{48 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {17 a \sqrt {d} e^{2} x^{5}}{24 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {5 a d^{3} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{16 \sqrt {e}} + \frac {a e^{3} x^{7}}{6 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {5 b d^{\frac {7}{2}} x}{128 e \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {133 b d^{\frac {5}{2}} x^{3}}{384 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {127 b d^{\frac {3}{2}} e x^{5}}{192 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {23 b \sqrt {d} e^{2} x^{7}}{48 \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {5 b d^{4} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{128 e^{\frac {3}{2}}} + \frac {b e^{3} x^{9}}{8 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {3 c d^{\frac {9}{2}} x}{256 e^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {c d^{\frac {7}{2}} x^{3}}{256 e \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {129 c d^{\frac {5}{2}} x^{5}}{640 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {73 c d^{\frac {3}{2}} e x^{7}}{160 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {29 c \sqrt {d} e^{2} x^{9}}{80 \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 c d^{5} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{256 e^{\frac {5}{2}}} + \frac {c e^{3} x^{11}}{10 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**(5/2)*x*sqrt(1 + e*x**2/d)/2 + 3*a*d**(5/2)*x/(16*sqrt(1 + e*x**2/d)) + 35*a*d**(3/2)*e*x**3/(48*sqrt(1 +
e*x**2/d)) + 17*a*sqrt(d)*e**2*x**5/(24*sqrt(1 + e*x**2/d)) + 5*a*d**3*asinh(sqrt(e)*x/sqrt(d))/(16*sqrt(e))
+ a*e**3*x**7/(6*sqrt(d)*sqrt(1 + e*x**2/d)) + 5*b*d**(7/2)*x/(128*e*sqrt(1 + e*x**2/d)) + 133*b*d**(5/2)*x**3
/(384*sqrt(1 + e*x**2/d)) + 127*b*d**(3/2)*e*x**5/(192*sqrt(1 + e*x**2/d)) + 23*b*sqrt(d)*e**2*x**7/(48*sqrt(1
+ e*x**2/d)) - 5*b*d**4*asinh(sqrt(e)*x/sqrt(d))/(128*e**(3/2)) + b*e**3*x**9/(8*sqrt(d)*sqrt(1 + e*x**2/d))
- 3*c*d**(9/2)*x/(256*e**2*sqrt(1 + e*x**2/d)) - c*d**(7/2)*x**3/(256*e*sqrt(1 + e*x**2/d)) + 129*c*d**(5/2)*x
**5/(640*sqrt(1 + e*x**2/d)) + 73*c*d**(3/2)*e*x**7/(160*sqrt(1 + e*x**2/d)) + 29*c*sqrt(d)*e**2*x**9/(80*sqrt
(1 + e*x**2/d)) + 3*c*d**5*asinh(sqrt(e)*x/sqrt(d))/(256*e**(5/2)) + c*e**3*x**11/(10*sqrt(d)*sqrt(1 + e*x**2/
d))

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