3.2.52 \(\int \frac {x^2 (c+d x^2+e x^4+f x^6)}{\sqrt {a+b x^2}} \, dx\) [152]

Optimal. Leaf size=194 \[ \frac {\left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) x \sqrt {a+b x^2}}{128 b^4}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}} \]

[Out]

-1/128*a*(-35*a^3*f+40*a^2*b*e-48*a*b^2*d+64*b^3*c)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(9/2)+1/128*(-35*a^3*
f+40*a^2*b*e-48*a*b^2*d+64*b^3*c)*x*(b*x^2+a)^(1/2)/b^4+1/192*(35*a^2*f-40*a*b*e+48*b^2*d)*x^3*(b*x^2+a)^(1/2)
/b^3+1/48*(-7*a*f+8*b*e)*x^5*(b*x^2+a)^(1/2)/b^2+1/8*f*x^7*(b*x^2+a)^(1/2)/b

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1823, 1281, 470, 327, 223, 212} \begin {gather*} \frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac {x \sqrt {a+b x^2} \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^4}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

((64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x*Sqrt[a + b*x^2])/(128*b^4) + ((48*b^2*d - 40*a*b*e + 35*a^2
*f)*x^3*Sqrt[a + b*x^2])/(192*b^3) + ((8*b*e - 7*a*f)*x^5*Sqrt[a + b*x^2])/(48*b^2) + (f*x^7*Sqrt[a + b*x^2])/
(8*b) - (a*(64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(9/2))

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 327

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

Rule 470

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

Rule 1281

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

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rubi steps

\begin {align*} \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx &=\frac {f x^7 \sqrt {a+b x^2}}{8 b}+\frac {\int \frac {x^2 \left (8 b c+8 b d x^2+(8 b e-7 a f) x^4\right )}{\sqrt {a+b x^2}} \, dx}{8 b}\\ &=\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}+\frac {\int \frac {x^2 \left (48 b^2 c+\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{48 b^2}\\ &=\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {1}{64} \left (-64 c+\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx\\ &=\frac {\left (64 c-\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt {a+b x^2}}{128 b}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {\left (a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^4}\\ &=\frac {\left (64 c-\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt {a+b x^2}}{128 b}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {\left (a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^4}\\ &=\frac {\left (64 c-\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt {a+b x^2}}{128 b}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.28, size = 151, normalized size = 0.78 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^3 f+10 a^2 b \left (12 e+7 f x^2\right )-8 a b^2 \left (18 d+10 e x^2+7 f x^4\right )+16 b^3 \left (12 c+6 d x^2+4 e x^4+3 f x^6\right )\right )-3 a \left (-64 b^3 c+48 a b^2 d-40 a^2 b e+35 a^3 f\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^3*f + 10*a^2*b*(12*e + 7*f*x^2) - 8*a*b^2*(18*d + 10*e*x^2 + 7*f*x^4) + 16*
b^3*(12*c + 6*d*x^2 + 4*e*x^4 + 3*f*x^6)) - 3*a*(-64*b^3*c + 48*a*b^2*d - 40*a^2*b*e + 35*a^3*f)*Log[-(Sqrt[b]
*x) + Sqrt[a + b*x^2]])/(384*b^(9/2))

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 306, normalized size = 1.58

method	result	size
risch	\(-\frac {x \left (-48 f \,x^{6} b^{3}+56 a \,b^{2} f \,x^{4}-64 b^{3} e \,x^{4}-70 a^{2} b f \,x^{2}+80 a \,b^{2} e \,x^{2}-96 b^{3} d \,x^{2}+105 a^{3} f -120 a^{2} b e +144 a \,b^{2} d -192 b^{3} c \right ) \sqrt {b \,x^{2}+a}}{384 b^{4}}+\frac {35 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) f}{128 b^{\frac {9}{2}}}-\frac {5 a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) e}{16 b^{\frac {7}{2}}}+\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) d}{8 b^{\frac {5}{2}}}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c}{2 b^{\frac {3}{2}}}\)	\(199\)
default	\(f \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )+e \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+d \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+c \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

f*(1/8*x^7/b*(b*x^2+a)^(1/2)-7/8*a/b*(1/6*x^5/b*(b*x^2+a)^(1/2)-5/6*a/b*(1/4*x^3/b*(b*x^2+a)^(1/2)-3/4*a/b*(1/
2*x*(b*x^2+a)^(1/2)/b-1/2*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))+e*(1/6*x^5/b*(b*x^2+a)^(1/2)-5/6*a/b*(1/
4*x^3/b*(b*x^2+a)^(1/2)-3/4*a/b*(1/2*x*(b*x^2+a)^(1/2)/b-1/2*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))))+d*(1/4
*x^3/b*(b*x^2+a)^(1/2)-3/4*a/b*(1/2*x*(b*x^2+a)^(1/2)/b-1/2*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))+c*(1/2*x
*(b*x^2+a)^(1/2)/b-1/2*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 259, normalized size = 1.34 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{7}}{8 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a f x^{5}}{48 \, b^{2}} + \frac {\sqrt {b x^{2} + a} x^{5} e}{6 \, b} + \frac {\sqrt {b x^{2} + a} d x^{3}}{4 \, b} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} f x^{3}}{192 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} a x^{3} e}{24 \, b^{2}} + \frac {\sqrt {b x^{2} + a} c x}{2 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a d x}{8 \, b^{2}} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} f x}{128 \, b^{4}} - \frac {a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {3 \, a^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {35 \, a^{4} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{2}}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} x e}{16 \, b^{3}} - \frac {5 \, a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) e}{16 \, b^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

1/8*sqrt(b*x^2 + a)*f*x^7/b - 7/48*sqrt(b*x^2 + a)*a*f*x^5/b^2 + 1/6*sqrt(b*x^2 + a)*x^5*e/b + 1/4*sqrt(b*x^2
+ a)*d*x^3/b + 35/192*sqrt(b*x^2 + a)*a^2*f*x^3/b^3 - 5/24*sqrt(b*x^2 + a)*a*x^3*e/b^2 + 1/2*sqrt(b*x^2 + a)*c
*x/b - 3/8*sqrt(b*x^2 + a)*a*d*x/b^2 - 35/128*sqrt(b*x^2 + a)*a^3*f*x/b^4 - 1/2*a*c*arcsinh(b*x/sqrt(a*b))/b^(
3/2) + 3/8*a^2*d*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 35/128*a^4*f*arcsinh(b*x/sqrt(a*b))/b^(9/2) + 5/16*sqrt(b*x^
2 + a)*a^2*x*e/b^3 - 5/16*a^3*arcsinh(b*x/sqrt(a*b))*e/b^(7/2)

________________________________________________________________________________________

Fricas [A]
time = 2.73, size = 343, normalized size = 1.77 \begin {gather*} \left [\frac {3 \, {\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d - 35 \, a^{4} f + 40 \, a^{3} b e\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (48 \, b^{4} f x^{7} - 56 \, a b^{3} f x^{5} + 2 \, {\left (48 \, b^{4} d + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \, {\left (64 \, b^{4} c - 48 \, a b^{3} d - 35 \, a^{3} b f\right )} x + 8 \, {\left (8 \, b^{4} x^{5} - 10 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x\right )} e\right )} \sqrt {b x^{2} + a}}{768 \, b^{5}}, \frac {3 \, {\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d - 35 \, a^{4} f + 40 \, a^{3} b e\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, b^{4} f x^{7} - 56 \, a b^{3} f x^{5} + 2 \, {\left (48 \, b^{4} d + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \, {\left (64 \, b^{4} c - 48 \, a b^{3} d - 35 \, a^{3} b f\right )} x + 8 \, {\left (8 \, b^{4} x^{5} - 10 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x\right )} e\right )} \sqrt {b x^{2} + a}}{384 \, b^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/768*(3*(64*a*b^3*c - 48*a^2*b^2*d - 35*a^4*f + 40*a^3*b*e)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)
*x - a) + 2*(48*b^4*f*x^7 - 56*a*b^3*f*x^5 + 2*(48*b^4*d + 35*a^2*b^2*f)*x^3 + 3*(64*b^4*c - 48*a*b^3*d - 35*a
^3*b*f)*x + 8*(8*b^4*x^5 - 10*a*b^3*x^3 + 15*a^2*b^2*x)*e)*sqrt(b*x^2 + a))/b^5, 1/384*(3*(64*a*b^3*c - 48*a^2
*b^2*d - 35*a^4*f + 40*a^3*b*e)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (48*b^4*f*x^7 - 56*a*b^3*f*x^5 +
2*(48*b^4*d + 35*a^2*b^2*f)*x^3 + 3*(64*b^4*c - 48*a*b^3*d - 35*a^3*b*f)*x + 8*(8*b^4*x^5 - 10*a*b^3*x^3 + 15
*a^2*b^2*x)*e)*sqrt(b*x^2 + a))/b^5]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (196) = 392\).
time = 47.78, size = 444, normalized size = 2.29 \begin {gather*} - \frac {35 a^{\frac {7}{2}} f x}{128 b^{4} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {5}{2}} e x}{16 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {35 a^{\frac {5}{2}} f x^{3}}{384 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {3}{2}} d x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {3}{2}} e x^{3}}{48 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 a^{\frac {3}{2}} f x^{5}}{192 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {\sqrt {a} c x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {\sqrt {a} d x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} e x^{5}}{24 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} f x^{7}}{48 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{4} f \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {9}{2}}} - \frac {5 a^{3} e \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {7}{2}}} + \frac {3 a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {a c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + \frac {d x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {e x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {f x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

-35*a**(7/2)*f*x/(128*b**4*sqrt(1 + b*x**2/a)) + 5*a**(5/2)*e*x/(16*b**3*sqrt(1 + b*x**2/a)) - 35*a**(5/2)*f*x
**3/(384*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*d*x/(8*b**2*sqrt(1 + b*x**2/a)) + 5*a**(3/2)*e*x**3/(48*b**2*sq
rt(1 + b*x**2/a)) + 7*a**(3/2)*f*x**5/(192*b**2*sqrt(1 + b*x**2/a)) + sqrt(a)*c*x*sqrt(1 + b*x**2/a)/(2*b) - s
qrt(a)*d*x**3/(8*b*sqrt(1 + b*x**2/a)) - sqrt(a)*e*x**5/(24*b*sqrt(1 + b*x**2/a)) - sqrt(a)*f*x**7/(48*b*sqrt(
1 + b*x**2/a)) + 35*a**4*f*asinh(sqrt(b)*x/sqrt(a))/(128*b**(9/2)) - 5*a**3*e*asinh(sqrt(b)*x/sqrt(a))/(16*b**
(7/2)) + 3*a**2*d*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) - a*c*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2)) + d*x**5/(
4*sqrt(a)*sqrt(1 + b*x**2/a)) + e*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + f*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

________________________________________________________________________________________

Giac [A]
time = 1.40, size = 175, normalized size = 0.90 \begin {gather*} \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, f x^{2}}{b} - \frac {7 \, a b^{5} f - 8 \, b^{6} e}{b^{7}}\right )} x^{2} + \frac {48 \, b^{6} d + 35 \, a^{2} b^{4} f - 40 \, a b^{5} e}{b^{7}}\right )} x^{2} + \frac {3 \, {\left (64 \, b^{6} c - 48 \, a b^{5} d - 35 \, a^{3} b^{3} f + 40 \, a^{2} b^{4} e\right )}}{b^{7}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d - 35 \, a^{4} f + 40 \, a^{3} b e\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*f*x^2/b - (7*a*b^5*f - 8*b^6*e)/b^7)*x^2 + (48*b^6*d + 35*a^2*b^4*f - 40*a*b^5*e)/b^7)*x^2 + 3*
(64*b^6*c - 48*a*b^5*d - 35*a^3*b^3*f + 40*a^2*b^4*e)/b^7)*sqrt(b*x^2 + a)*x + 1/128*(64*a*b^3*c - 48*a^2*b^2*
d - 35*a^4*f + 40*a^3*b*e)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (f\,x^6+e\,x^4+d\,x^2+c\right )}{\sqrt {b\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^(1/2),x)

[Out]

int((x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^(1/2), x)

________________________________________________________________________________________