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

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

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Rubi [A]  time = 0.26, antiderivative size = 245, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.24, size = 184, normalized size = 0.75 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (945 a^4 f-210 a^3 b \left (5 e+3 f x^2\right )+4 a^2 b^2 \left (300 d+175 e x^2+126 f x^4\right )-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 (63 a^3 f-70 a^2 b e+80 a b^2 d-96 b^3 c\right )}{3840 b^{11/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.44, size = 215, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x^2} \left (945 a^4 f x-1050 a^3 b e x-630 a^3 b f x^3+1200 a^2 b^2 d x+700 a^2 b^2 e x^3+504 a^2 b^2 f x^5-1440 a b^3 c x-800 a b^3 d x^3-560 a b^3 e x^5-432 a b^3 f x^7+960 b^4 c x^3+640 b^4 d x^5+480 b^4 e x^7+384 b^4 f x^9\right )}{3840 b^5}+\frac {\log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \left (63 a^5 f-70 a^4 b e+80 a^3 b^2 d-96 a^2 b^3 c\right )}{256 b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^2]*(-1440*a*b^3*c*x + 1200*a^2*b^2*d*x - 1050*a^3*b*e*x + 945*a^4*f*x + 960*b^4*c*x^3 - 800*a*b^
3*d*x^3 + 700*a^2*b^2*e*x^3 - 630*a^3*b*f*x^3 + 640*b^4*d*x^5 - 560*a*b^3*e*x^5 + 504*a^2*b^2*f*x^5 + 480*b^4*
e*x^7 - 432*a*b^3*f*x^7 + 384*b^4*f*x^9))/(3840*b^5) + ((-96*a^2*b^3*c + 80*a^3*b^2*d - 70*a^4*b*e + 63*a^5*f)
*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(256*b^(11/2))

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fricas [A]  time = 1.44, size = 414, normalized size = 1.69 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.53, size = 224, normalized size = 0.91 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.02, size = 368, normalized size = 1.50 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, f \,x^{9}}{10 b}-\frac {9 \sqrt {b \,x^{2}+a}\, a f \,x^{7}}{80 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, e \,x^{7}}{8 b}+\frac {21 \sqrt {b \,x^{2}+a}\, a^{2} f \,x^{5}}{160 b^{3}}-\frac {7 \sqrt {b \,x^{2}+a}\, a e \,x^{5}}{48 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, d \,x^{5}}{6 b}-\frac {21 \sqrt {b \,x^{2}+a}\, a^{3} f \,x^{3}}{128 b^{4}}+\frac {35 \sqrt {b \,x^{2}+a}\, a^{2} e \,x^{3}}{192 b^{3}}-\frac {5 \sqrt {b \,x^{2}+a}\, a d \,x^{3}}{24 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, c \,x^{3}}{4 b}-\frac {63 a^{5} f \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {11}{2}}}+\frac {35 a^{4} e \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {9}{2}}}-\frac {5 a^{3} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {7}{2}}}+\frac {3 a^{2} c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}+\frac {63 \sqrt {b \,x^{2}+a}\, a^{4} f x}{256 b^{5}}-\frac {35 \sqrt {b \,x^{2}+a}\, a^{3} e x}{128 b^{4}}+\frac {5 \sqrt {b \,x^{2}+a}\, a^{2} d x}{16 b^{3}}-\frac {3 \sqrt {b \,x^{2}+a}\, a c x}{8 b^{2}} \end {gather*}

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

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maxima [A]  time = 1.46, size = 339, normalized size = 1.38 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{9}}{10 \, b} + \frac {\sqrt {b x^{2} + a} e x^{7}}{8 \, b} - \frac {9 \, \sqrt {b x^{2} + a} a f x^{7}}{80 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{5}}{6 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a e x^{5}}{48 \, b^{2}} + \frac {21 \, \sqrt {b x^{2} + a} a^{2} f x^{5}}{160 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a d x^{3}}{24 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} e x^{3}}{192 \, b^{3}} - \frac {21 \, \sqrt {b x^{2} + a} a^{3} f x^{3}}{128 \, b^{4}} - \frac {3 \, \sqrt {b x^{2} + a} a c x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} d x}{16 \, b^{3}} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} e x}{128 \, b^{4}} + \frac {63 \, \sqrt {b x^{2} + a} a^{4} f x}{256 \, b^{5}} + \frac {3 \, a^{2} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} + \frac {35 \, a^{4} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{2}}} - \frac {63 \, a^{5} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {11}{2}}} \end {gather*}

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]

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

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

[Out]

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

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sympy [B]  time = 42.12, size = 586, normalized size = 2.39 \begin {gather*} \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 {gather*}

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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