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

Optimal. Leaf size=245 \[ \frac{x^3 \sqrt{a+b x^2} \left (70 a^2 b e-63 a^3 f-80 a b^2 d+96 b^3 c\right )}{384 b^4}-\frac{a x \sqrt{a+b x^2} \left (70 a^2 b e-63 a^3 f-80 a b^2 d+96 b^3 c\right )}{256 b^5}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (70 a^2 b e-63 a^3 f-80 a b^2 d+96 b^3 c\right )}{256 b^{11/2}}+\frac{x^5 \sqrt{a+b x^2} \left (63 a^2 f-70 a b e+80 b^2 d\right )}{480 b^3}+\frac{x^7 \sqrt{a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b} \]

[Out]

-(a*(96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*x*Sqrt[a + b*x^2])/(256*b^5) + ((96*b^3*c - 80*a*b^2*d + 7
0*a^2*b*e - 63*a^3*f)*x^3*Sqrt[a + b*x^2])/(384*b^4) + ((80*b^2*d - 70*a*b*e + 63*a^2*f)*x^5*Sqrt[a + b*x^2])/
(480*b^3) + ((10*b*e - 9*a*f)*x^7*Sqrt[a + b*x^2])/(80*b^2) + (f*x^9*Sqrt[a + b*x^2])/(10*b) + (a^2*(96*b^3*c
- 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(11/2))

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Rubi [A]  time = 0.258489, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1809, 1267, 459, 321, 217, 206} \[ \frac{x^3 \sqrt{a+b x^2} \left (70 a^2 b e-63 a^3 f-80 a b^2 d+96 b^3 c\right )}{384 b^4}-\frac{a x \sqrt{a+b x^2} \left (70 a^2 b e-63 a^3 f-80 a b^2 d+96 b^3 c\right )}{256 b^5}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (70 a^2 b e-63 a^3 f-80 a b^2 d+96 b^3 c\right )}{256 b^{11/2}}+\frac{x^5 \sqrt{a+b x^2} \left (63 a^2 f-70 a b e+80 b^2 d\right )}{480 b^3}+\frac{x^7 \sqrt{a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*x*Sqrt[a + b*x^2])/(256*b^5) + ((96*b^3*c - 80*a*b^2*d + 7
0*a^2*b*e - 63*a^3*f)*x^3*Sqrt[a + b*x^2])/(384*b^4) + ((80*b^2*d - 70*a*b*e + 63*a^2*f)*x^5*Sqrt[a + b*x^2])/
(480*b^3) + ((10*b*e - 9*a*f)*x^7*Sqrt[a + b*x^2])/(80*b^2) + (f*x^9*Sqrt[a + b*x^2])/(10*b) + (a^2*(96*b^3*c
- 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(11/2))

Rule 1809

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])

Rule 1267

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 459

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 321

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 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 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])

Rubi steps

\begin{align*} \int \frac{x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx &=\frac{f x^9 \sqrt{a+b x^2}}{10 b}+\frac{\int \frac{x^4 \left (10 b c+10 b d x^2+(10 b e-9 a f) x^4\right )}{\sqrt{a+b x^2}} \, dx}{10 b}\\ &=\frac{(10 b e-9 a f) x^7 \sqrt{a+b x^2}}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}+\frac{\int \frac{x^4 \left (80 b^2 c+\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^2\right )}{\sqrt{a+b x^2}} \, dx}{80 b^2}\\ &=\frac{\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt{a+b x^2}}{480 b^3}+\frac{(10 b e-9 a f) x^7 \sqrt{a+b x^2}}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}+\frac{\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \int \frac{x^4}{\sqrt{a+b x^2}} \, dx}{96 b^3}\\ &=\frac{\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt{a+b x^2}}{384 b^4}+\frac{\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt{a+b x^2}}{480 b^3}+\frac{(10 b e-9 a f) x^7 \sqrt{a+b x^2}}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}-\frac{\left (a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{128 b^4}\\ &=-\frac{a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt{a+b x^2}}{256 b^5}+\frac{\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt{a+b x^2}}{384 b^4}+\frac{\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt{a+b x^2}}{480 b^3}+\frac{(10 b e-9 a f) x^7 \sqrt{a+b x^2}}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}+\frac{\left (a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{256 b^5}\\ &=-\frac{a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt{a+b x^2}}{256 b^5}+\frac{\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt{a+b x^2}}{384 b^4}+\frac{\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt{a+b x^2}}{480 b^3}+\frac{(10 b e-9 a f) x^7 \sqrt{a+b x^2}}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}+\frac{\left (a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{256 b^5}\\ &=-\frac{a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt{a+b x^2}}{256 b^5}+\frac{\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt{a+b x^2}}{384 b^4}+\frac{\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt{a+b x^2}}{480 b^3}+\frac{(10 b e-9 a f) x^7 \sqrt{a+b x^2}}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b}+\frac{a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.223898, size = 184, normalized size = 0.75 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (4 a^2 b^2 \left (300 d+175 e x^2+126 f x^4\right )-210 a^3 b \left (5 e+3 f x^2\right )+945 a^4 f-16 a b^3 \left (90 c+50 d x^2+35 e x^4+27 f x^6\right )+32 b^4 x^2 \left (30 c+20 d x^2+15 e x^4+12 f x^6\right )\right )-15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-70 a^2 b e+63 a^3 f+80 a b^2 d-96 b^3 c\right )}{3840 b^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(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]*(945*a^4*f - 210*a^3*b*(5*e + 3*f*x^2) + 4*a^2*b^2*(300*d + 175*e*x^2 + 126*f*x^4)
+ 32*b^4*x^2*(30*c + 20*d*x^2 + 15*e*x^4 + 12*f*x^6) - 16*a*b^3*(90*c + 50*d*x^2 + 35*e*x^4 + 27*f*x^6)) - 15*
a^2*(-96*b^3*c + 80*a*b^2*d - 70*a^2*b*e + 63*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(3840*b^(11/2))

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Maple [A]  time = 0.022, size = 368, normalized size = 1.5 \begin{align*}{\frac{f{x}^{9}}{10\,b}\sqrt{b{x}^{2}+a}}-{\frac{9\,af{x}^{7}}{80\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{21\,{a}^{2}f{x}^{5}}{160\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{21\,{a}^{3}f{x}^{3}}{128\,{b}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{63\,f{a}^{4}x}{256\,{b}^{5}}\sqrt{b{x}^{2}+a}}-{\frac{63\,f{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}}+{\frac{e{x}^{7}}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{7\,ae{x}^{5}}{48\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{35\,{a}^{2}e{x}^{3}}{192\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{35\,e{a}^{3}x}{128\,{b}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{35\,e{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{d{x}^{5}}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,ad{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{2}dx}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{a}^{3}d}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{c{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,acx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}c}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*f*x^9*(b*x^2+a)^(1/2)/b-9/80*f/b^2*a*x^7*(b*x^2+a)^(1/2)+21/160*f/b^3*a^2*x^5*(b*x^2+a)^(1/2)-21/128*f/b^
4*a^3*x^3*(b*x^2+a)^(1/2)+63/256*f/b^5*a^4*x*(b*x^2+a)^(1/2)-63/256*f/b^(11/2)*a^5*ln(x*b^(1/2)+(b*x^2+a)^(1/2
))+1/8*e*x^7/b*(b*x^2+a)^(1/2)-7/48*e/b^2*a*x^5*(b*x^2+a)^(1/2)+35/192*e/b^3*a^2*x^3*(b*x^2+a)^(1/2)-35/128*e/
b^4*a^3*x*(b*x^2+a)^(1/2)+35/128*e/b^(9/2)*a^4*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/6*d*x^5/b*(b*x^2+a)^(1/2)-5/24*
d/b^2*a*x^3*(b*x^2+a)^(1/2)+5/16*d/b^3*a^2*x*(b*x^2+a)^(1/2)-5/16*d/b^(7/2)*a^3*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+
1/4*c*x^3/b*(b*x^2+a)^(1/2)-3/8*c/b^2*a*x*(b*x^2+a)^(1/2)+3/8*c/b^(5/2)*a^2*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0697, size = 983, normalized size = 4.01 \begin{align*} \left [-\frac{15 \,{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (384 \, b^{5} f x^{9} + 48 \,{\left (10 \, b^{5} e - 9 \, a b^{4} f\right )} x^{7} + 8 \,{\left (80 \, b^{5} d - 70 \, a b^{4} e + 63 \, a^{2} b^{3} f\right )} x^{5} + 10 \,{\left (96 \, b^{5} c - 80 \, a b^{4} d + 70 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{3} - 15 \,{\left (96 \, a b^{4} c - 80 \, a^{2} b^{3} d + 70 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x\right )} \sqrt{b x^{2} + a}}{7680 \, b^{6}}, -\frac{15 \,{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (384 \, b^{5} f x^{9} + 48 \,{\left (10 \, b^{5} e - 9 \, a b^{4} f\right )} x^{7} + 8 \,{\left (80 \, b^{5} d - 70 \, a b^{4} e + 63 \, a^{2} b^{3} f\right )} x^{5} + 10 \,{\left (96 \, b^{5} c - 80 \, a b^{4} d + 70 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{3} - 15 \,{\left (96 \, a b^{4} c - 80 \, a^{2} b^{3} d + 70 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x\right )} \sqrt{b x^{2} + a}}{3840 \, b^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(96*a^2*b^3*c - 80*a^3*b^2*d + 70*a^4*b*e - 63*a^5*f)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sq
rt(b)*x - a) - 2*(384*b^5*f*x^9 + 48*(10*b^5*e - 9*a*b^4*f)*x^7 + 8*(80*b^5*d - 70*a*b^4*e + 63*a^2*b^3*f)*x^5
+ 10*(96*b^5*c - 80*a*b^4*d + 70*a^2*b^3*e - 63*a^3*b^2*f)*x^3 - 15*(96*a*b^4*c - 80*a^2*b^3*d + 70*a^3*b^2*e
- 63*a^4*b*f)*x)*sqrt(b*x^2 + a))/b^6, -1/3840*(15*(96*a^2*b^3*c - 80*a^3*b^2*d + 70*a^4*b*e - 63*a^5*f)*sqrt
(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (384*b^5*f*x^9 + 48*(10*b^5*e - 9*a*b^4*f)*x^7 + 8*(80*b^5*d - 70*a*
b^4*e + 63*a^2*b^3*f)*x^5 + 10*(96*b^5*c - 80*a*b^4*d + 70*a^2*b^3*e - 63*a^3*b^2*f)*x^3 - 15*(96*a*b^4*c - 80
*a^2*b^3*d + 70*a^3*b^2*e - 63*a^4*b*f)*x)*sqrt(b*x^2 + a))/b^6]

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Sympy [B]  time = 33.212, size = 586, normalized size = 2.39 \begin{align*} \frac{63 a^{\frac{9}{2}} f x}{256 b^{5} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{\frac{7}{2}} e x}{128 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{21 a^{\frac{7}{2}} f x^{3}}{256 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{5}{2}} d x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{\frac{5}{2}} e x^{3}}{384 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{21 a^{\frac{5}{2}} f x^{5}}{640 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{3}{2}} c x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{3}{2}} d x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{7 a^{\frac{3}{2}} e x^{5}}{192 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{\frac{3}{2}} f x^{7}}{160 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} c x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} d x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} e x^{7}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} f x^{9}}{80 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{63 a^{5} f \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{11}{2}}} + \frac{35 a^{4} e \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{9}{2}}} - \frac{5 a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{3 a^{2} c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{c x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{d x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{e x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{f x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

63*a**(9/2)*f*x/(256*b**5*sqrt(1 + b*x**2/a)) - 35*a**(7/2)*e*x/(128*b**4*sqrt(1 + b*x**2/a)) + 21*a**(7/2)*f*
x**3/(256*b**4*sqrt(1 + b*x**2/a)) + 5*a**(5/2)*d*x/(16*b**3*sqrt(1 + b*x**2/a)) - 35*a**(5/2)*e*x**3/(384*b**
3*sqrt(1 + b*x**2/a)) - 21*a**(5/2)*f*x**5/(640*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*c*x/(8*b**2*sqrt(1 + b*x
**2/a)) + 5*a**(3/2)*d*x**3/(48*b**2*sqrt(1 + b*x**2/a)) + 7*a**(3/2)*e*x**5/(192*b**2*sqrt(1 + b*x**2/a)) + 3
*a**(3/2)*f*x**7/(160*b**2*sqrt(1 + b*x**2/a)) - sqrt(a)*c*x**3/(8*b*sqrt(1 + b*x**2/a)) - sqrt(a)*d*x**5/(24*
b*sqrt(1 + b*x**2/a)) - sqrt(a)*e*x**7/(48*b*sqrt(1 + b*x**2/a)) - sqrt(a)*f*x**9/(80*b*sqrt(1 + b*x**2/a)) -
63*a**5*f*asinh(sqrt(b)*x/sqrt(a))/(256*b**(11/2)) + 35*a**4*e*asinh(sqrt(b)*x/sqrt(a))/(128*b**(9/2)) - 5*a**
3*d*asinh(sqrt(b)*x/sqrt(a))/(16*b**(7/2)) + 3*a**2*c*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) + c*x**5/(4*sqrt(a
)*sqrt(1 + b*x**2/a)) + d*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + e*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a)) + f*x**1
1/(10*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.18716, size = 302, normalized size = 1.23 \begin{align*} \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, f x^{2}}{b} - \frac{9 \, a b^{7} f - 10 \, b^{8} e}{b^{9}}\right )} x^{2} + \frac{80 \, b^{8} d + 63 \, a^{2} b^{6} f - 70 \, a b^{7} e}{b^{9}}\right )} x^{2} + \frac{5 \,{\left (96 \, b^{8} c - 80 \, a b^{7} d - 63 \, a^{3} b^{5} f + 70 \, a^{2} b^{6} e\right )}}{b^{9}}\right )} x^{2} - \frac{15 \,{\left (96 \, a b^{7} c - 80 \, a^{2} b^{6} d - 63 \, a^{4} b^{4} f + 70 \, a^{3} b^{5} e\right )}}{b^{9}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d - 63 \, a^{5} f + 70 \, a^{4} b e\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(2*(4*(6*(8*f*x^2/b - (9*a*b^7*f - 10*b^8*e)/b^9)*x^2 + (80*b^8*d + 63*a^2*b^6*f - 70*a*b^7*e)/b^9)*x^2
+ 5*(96*b^8*c - 80*a*b^7*d - 63*a^3*b^5*f + 70*a^2*b^6*e)/b^9)*x^2 - 15*(96*a*b^7*c - 80*a^2*b^6*d - 63*a^4*b
^4*f + 70*a^3*b^5*e)/b^9)*sqrt(b*x^2 + a)*x - 1/256*(96*a^2*b^3*c - 80*a^3*b^2*d - 63*a^5*f + 70*a^4*b*e)*log(
abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)