3.169 \(\int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx\)

Optimal. Leaf size=214 \[ \frac {\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac {\left (a+b x^2\right )^{5/2} \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac {a \left (a+b x^2\right )^{3/2} \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac {a^2 \sqrt {a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}+\frac {\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac {f \left (a+b x^2\right )^{11/2}}{11 b^6} \]

[Out]

-1/3*a*(-5*a^3*f+4*a^2*b*e-3*a*b^2*d+2*b^3*c)*(b*x^2+a)^(3/2)/b^6+1/5*(-10*a^3*f+6*a^2*b*e-3*a*b^2*d+b^3*c)*(b
*x^2+a)^(5/2)/b^6+1/7*(10*a^2*f-4*a*b*e+b^2*d)*(b*x^2+a)^(7/2)/b^6+1/9*(-5*a*f+b*e)*(b*x^2+a)^(9/2)/b^6+1/11*f
*(b*x^2+a)^(11/2)/b^6+a^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*(b*x^2+a)^(1/2)/b^6

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Rubi [A]  time = 0.22, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1811, 1799, 1850} \[ \frac {\left (a+b x^2\right )^{5/2} \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac {a \left (a+b x^2\right )^{3/2} \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac {a^2 \sqrt {a+b x^2} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}+\frac {\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac {\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac {f \left (a+b x^2\right )^{11/2}}{11 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^5 + d*x^7 + e*x^9 + f*x^11)/Sqrt[a + b*x^2],x]

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^6 - (a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)
*(a + b*x^2)^(3/2))/(3*b^6) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*(a + b*x^2)^(5/2))/(5*b^6) + ((b^2*d
- 4*a*b*e + 10*a^2*f)*(a + b*x^2)^(7/2))/(7*b^6) + ((b*e - 5*a*f)*(a + b*x^2)^(9/2))/(9*b^6) + (f*(a + b*x^2)
^(11/2))/(11*b^6)

Rule 1799

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

Rule 1811

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x] /; Fre
eQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {align*} \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx &=\int \frac {x \left (c x^4+d x^6+e x^8+f x^{10}\right )}{\sqrt {a+b x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {c x^2+d x^3+e x^4+f x^5}{\sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 \sqrt {a+b x}}+\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) \sqrt {a+b x}}{b^5}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) (a+b x)^{3/2}}{b^5}+\frac {\left (b^2 d-4 a b e+10 a^2 f\right ) (a+b x)^{5/2}}{b^5}+\frac {(b e-5 a f) (a+b x)^{7/2}}{b^5}+\frac {f (a+b x)^{9/2}}{b^5}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt {a+b x^2}}{b^6}-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) \left (a+b x^2\right )^{3/2}}{3 b^6}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) \left (a+b x^2\right )^{5/2}}{5 b^6}+\frac {\left (b^2 d-4 a b e+10 a^2 f\right ) \left (a+b x^2\right )^{7/2}}{7 b^6}+\frac {(b e-5 a f) \left (a+b x^2\right )^{9/2}}{9 b^6}+\frac {f \left (a+b x^2\right )^{11/2}}{11 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 158, normalized size = 0.74 \[ \frac {\sqrt {a+b x^2} \left (-1280 a^5 f+128 a^4 b \left (11 e+5 f x^2\right )-16 a^3 b^2 \left (99 d+44 e x^2+30 f x^4\right )+8 a^2 b^3 \left (231 c+99 d x^2+66 e x^4+50 f x^6\right )-2 a b^4 x^2 \left (462 c+297 d x^2+220 e x^4+175 f x^6\right )+b^5 x^4 \left (693 c+5 \left (99 d x^2+77 e x^4+63 f x^6\right )\right )\right )}{3465 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^5 + d*x^7 + e*x^9 + f*x^11)/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(-1280*a^5*f + 128*a^4*b*(11*e + 5*f*x^2) - 16*a^3*b^2*(99*d + 44*e*x^2 + 30*f*x^4) + 8*a^2*b
^3*(231*c + 99*d*x^2 + 66*e*x^4 + 50*f*x^6) - 2*a*b^4*x^2*(462*c + 297*d*x^2 + 220*e*x^4 + 175*f*x^6) + b^5*x^
4*(693*c + 5*(99*d*x^2 + 77*e*x^4 + 63*f*x^6))))/(3465*b^6)

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fricas [A]  time = 0.55, size = 177, normalized size = 0.83 \[ \frac {{\left (315 \, b^{5} f x^{10} + 35 \, {\left (11 \, b^{5} e - 10 \, a b^{4} f\right )} x^{8} + 5 \, {\left (99 \, b^{5} d - 88 \, a b^{4} e + 80 \, a^{2} b^{3} f\right )} x^{6} + 1848 \, a^{2} b^{3} c - 1584 \, a^{3} b^{2} d + 1408 \, a^{4} b e - 1280 \, a^{5} f + 3 \, {\left (231 \, b^{5} c - 198 \, a b^{4} d + 176 \, a^{2} b^{3} e - 160 \, a^{3} b^{2} f\right )} x^{4} - 4 \, {\left (231 \, a b^{4} c - 198 \, a^{2} b^{3} d + 176 \, a^{3} b^{2} e - 160 \, a^{4} b f\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^11+e*x^9+d*x^7+c*x^5)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/3465*(315*b^5*f*x^10 + 35*(11*b^5*e - 10*a*b^4*f)*x^8 + 5*(99*b^5*d - 88*a*b^4*e + 80*a^2*b^3*f)*x^6 + 1848*
a^2*b^3*c - 1584*a^3*b^2*d + 1408*a^4*b*e - 1280*a^5*f + 3*(231*b^5*c - 198*a*b^4*d + 176*a^2*b^3*e - 160*a^3*
b^2*f)*x^4 - 4*(231*a*b^4*c - 198*a^2*b^3*d + 176*a^3*b^2*e - 160*a^4*b*f)*x^2)*sqrt(b*x^2 + a)/b^6

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giac [A]  time = 0.45, size = 264, normalized size = 1.23 \[ \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )} \sqrt {b x^{2} + a}}{b^{6}} + \frac {693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} c - 2310 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{3} c + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2} d - 2079 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b^{2} d + 3465 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b^{2} d + 315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} f - 1925 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a f + 4950 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} f - 6930 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} f + 5775 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} f + 385 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b e - 1980 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b e + 4158 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b e - 4620 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b e}{3465 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^11+e*x^9+d*x^7+c*x^5)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

(a^2*b^3*c - a^3*b^2*d - a^5*f + a^4*b*e)*sqrt(b*x^2 + a)/b^6 + 1/3465*(693*(b*x^2 + a)^(5/2)*b^3*c - 2310*(b*
x^2 + a)^(3/2)*a*b^3*c + 495*(b*x^2 + a)^(7/2)*b^2*d - 2079*(b*x^2 + a)^(5/2)*a*b^2*d + 3465*(b*x^2 + a)^(3/2)
*a^2*b^2*d + 315*(b*x^2 + a)^(11/2)*f - 1925*(b*x^2 + a)^(9/2)*a*f + 4950*(b*x^2 + a)^(7/2)*a^2*f - 6930*(b*x^
2 + a)^(5/2)*a^3*f + 5775*(b*x^2 + a)^(3/2)*a^4*f + 385*(b*x^2 + a)^(9/2)*b*e - 1980*(b*x^2 + a)^(7/2)*a*b*e +
4158*(b*x^2 + a)^(5/2)*a^2*b*e - 4620*(b*x^2 + a)^(3/2)*a^3*b*e)/b^6

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maple [A]  time = 0.01, size = 193, normalized size = 0.90 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-315 f \,x^{10} b^{5}+350 a \,b^{4} f \,x^{8}-385 b^{5} e \,x^{8}-400 a^{2} b^{3} f \,x^{6}+440 a \,b^{4} e \,x^{6}-495 b^{5} d \,x^{6}+480 a^{3} b^{2} f \,x^{4}-528 a^{2} b^{3} e \,x^{4}+594 a \,b^{4} d \,x^{4}-693 b^{5} c \,x^{4}-640 a^{4} b f \,x^{2}+704 a^{3} b^{2} e \,x^{2}-792 a^{2} b^{3} d \,x^{2}+924 a \,b^{4} c \,x^{2}+1280 a^{5} f -1408 a^{4} b e +1584 a^{3} b^{2} d -1848 a^{2} b^{3} c \right )}{3465 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^11+e*x^9+d*x^7+c*x^5)/(b*x^2+a)^(1/2),x)

[Out]

-1/3465*(b*x^2+a)^(1/2)*(-315*b^5*f*x^10+350*a*b^4*f*x^8-385*b^5*e*x^8-400*a^2*b^3*f*x^6+440*a*b^4*e*x^6-495*b
^5*d*x^6+480*a^3*b^2*f*x^4-528*a^2*b^3*e*x^4+594*a*b^4*d*x^4-693*b^5*c*x^4-640*a^4*b*f*x^2+704*a^3*b^2*e*x^2-7
92*a^2*b^3*d*x^2+924*a*b^4*c*x^2+1280*a^5*f-1408*a^4*b*e+1584*a^3*b^2*d-1848*a^2*b^3*c)/b^6

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maxima [A]  time = 1.43, size = 347, normalized size = 1.62 \[ \frac {\sqrt {b x^{2} + a} f x^{10}}{11 \, b} + \frac {\sqrt {b x^{2} + a} e x^{8}}{9 \, b} - \frac {10 \, \sqrt {b x^{2} + a} a f x^{8}}{99 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{6}}{7 \, b} - \frac {8 \, \sqrt {b x^{2} + a} a e x^{6}}{63 \, b^{2}} + \frac {80 \, \sqrt {b x^{2} + a} a^{2} f x^{6}}{693 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x^{4}}{5 \, b} - \frac {6 \, \sqrt {b x^{2} + a} a d x^{4}}{35 \, b^{2}} + \frac {16 \, \sqrt {b x^{2} + a} a^{2} e x^{4}}{105 \, b^{3}} - \frac {32 \, \sqrt {b x^{2} + a} a^{3} f x^{4}}{231 \, b^{4}} - \frac {4 \, \sqrt {b x^{2} + a} a c x^{2}}{15 \, b^{2}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} d x^{2}}{35 \, b^{3}} - \frac {64 \, \sqrt {b x^{2} + a} a^{3} e x^{2}}{315 \, b^{4}} + \frac {128 \, \sqrt {b x^{2} + a} a^{4} f x^{2}}{693 \, b^{5}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} c}{15 \, b^{3}} - \frac {16 \, \sqrt {b x^{2} + a} a^{3} d}{35 \, b^{4}} + \frac {128 \, \sqrt {b x^{2} + a} a^{4} e}{315 \, b^{5}} - \frac {256 \, \sqrt {b x^{2} + a} a^{5} f}{693 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^11+e*x^9+d*x^7+c*x^5)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

1/11*sqrt(b*x^2 + a)*f*x^10/b + 1/9*sqrt(b*x^2 + a)*e*x^8/b - 10/99*sqrt(b*x^2 + a)*a*f*x^8/b^2 + 1/7*sqrt(b*x
^2 + a)*d*x^6/b - 8/63*sqrt(b*x^2 + a)*a*e*x^6/b^2 + 80/693*sqrt(b*x^2 + a)*a^2*f*x^6/b^3 + 1/5*sqrt(b*x^2 + a
)*c*x^4/b - 6/35*sqrt(b*x^2 + a)*a*d*x^4/b^2 + 16/105*sqrt(b*x^2 + a)*a^2*e*x^4/b^3 - 32/231*sqrt(b*x^2 + a)*a
^3*f*x^4/b^4 - 4/15*sqrt(b*x^2 + a)*a*c*x^2/b^2 + 8/35*sqrt(b*x^2 + a)*a^2*d*x^2/b^3 - 64/315*sqrt(b*x^2 + a)*
a^3*e*x^2/b^4 + 128/693*sqrt(b*x^2 + a)*a^4*f*x^2/b^5 + 8/15*sqrt(b*x^2 + a)*a^2*c/b^3 - 16/35*sqrt(b*x^2 + a)
*a^3*d/b^4 + 128/315*sqrt(b*x^2 + a)*a^4*e/b^5 - 256/693*sqrt(b*x^2 + a)*a^5*f/b^6

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mupad [B]  time = 1.20, size = 186, normalized size = 0.87 \[ \sqrt {b\,x^2+a}\,\left (\frac {x^6\,\left (400\,f\,a^2\,b^3-440\,e\,a\,b^4+495\,d\,b^5\right )}{3465\,b^6}-\frac {1280\,f\,a^5-1408\,e\,a^4\,b+1584\,d\,a^3\,b^2-1848\,c\,a^2\,b^3}{3465\,b^6}+\frac {x^4\,\left (-480\,f\,a^3\,b^2+528\,e\,a^2\,b^3-594\,d\,a\,b^4+693\,c\,b^5\right )}{3465\,b^6}+\frac {f\,x^{10}}{11\,b}+\frac {x^8\,\left (385\,b^5\,e-350\,a\,b^4\,f\right )}{3465\,b^6}-\frac {4\,a\,x^2\,\left (-160\,f\,a^3+176\,e\,a^2\,b-198\,d\,a\,b^2+231\,c\,b^3\right )}{3465\,b^5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^5 + d*x^7 + e*x^9 + f*x^11)/(a + b*x^2)^(1/2),x)

[Out]

(a + b*x^2)^(1/2)*((x^6*(495*b^5*d + 400*a^2*b^3*f - 440*a*b^4*e))/(3465*b^6) - (1280*a^5*f - 1848*a^2*b^3*c +
1584*a^3*b^2*d - 1408*a^4*b*e)/(3465*b^6) + (x^4*(693*b^5*c + 528*a^2*b^3*e - 480*a^3*b^2*f - 594*a*b^4*d))/(
3465*b^6) + (f*x^10)/(11*b) + (x^8*(385*b^5*e - 350*a*b^4*f))/(3465*b^6) - (4*a*x^2*(231*b^3*c - 160*a^3*f - 1
98*a*b^2*d + 176*a^2*b*e))/(3465*b^5))

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sympy [A]  time = 9.37, size = 442, normalized size = 2.07 \[ \begin {cases} - \frac {256 a^{5} f \sqrt {a + b x^{2}}}{693 b^{6}} + \frac {128 a^{4} e \sqrt {a + b x^{2}}}{315 b^{5}} + \frac {128 a^{4} f x^{2} \sqrt {a + b x^{2}}}{693 b^{5}} - \frac {16 a^{3} d \sqrt {a + b x^{2}}}{35 b^{4}} - \frac {64 a^{3} e x^{2} \sqrt {a + b x^{2}}}{315 b^{4}} - \frac {32 a^{3} f x^{4} \sqrt {a + b x^{2}}}{231 b^{4}} + \frac {8 a^{2} c \sqrt {a + b x^{2}}}{15 b^{3}} + \frac {8 a^{2} d x^{2} \sqrt {a + b x^{2}}}{35 b^{3}} + \frac {16 a^{2} e x^{4} \sqrt {a + b x^{2}}}{105 b^{3}} + \frac {80 a^{2} f x^{6} \sqrt {a + b x^{2}}}{693 b^{3}} - \frac {4 a c x^{2} \sqrt {a + b x^{2}}}{15 b^{2}} - \frac {6 a d x^{4} \sqrt {a + b x^{2}}}{35 b^{2}} - \frac {8 a e x^{6} \sqrt {a + b x^{2}}}{63 b^{2}} - \frac {10 a f x^{8} \sqrt {a + b x^{2}}}{99 b^{2}} + \frac {c x^{4} \sqrt {a + b x^{2}}}{5 b} + \frac {d x^{6} \sqrt {a + b x^{2}}}{7 b} + \frac {e x^{8} \sqrt {a + b x^{2}}}{9 b} + \frac {f x^{10} \sqrt {a + b x^{2}}}{11 b} & \text {for}\: b \neq 0 \\\frac {\frac {c x^{6}}{6} + \frac {d x^{8}}{8} + \frac {e x^{10}}{10} + \frac {f x^{12}}{12}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**11+e*x**9+d*x**7+c*x**5)/(b*x**2+a)**(1/2),x)

[Out]

Piecewise((-256*a**5*f*sqrt(a + b*x**2)/(693*b**6) + 128*a**4*e*sqrt(a + b*x**2)/(315*b**5) + 128*a**4*f*x**2*
sqrt(a + b*x**2)/(693*b**5) - 16*a**3*d*sqrt(a + b*x**2)/(35*b**4) - 64*a**3*e*x**2*sqrt(a + b*x**2)/(315*b**4
) - 32*a**3*f*x**4*sqrt(a + b*x**2)/(231*b**4) + 8*a**2*c*sqrt(a + b*x**2)/(15*b**3) + 8*a**2*d*x**2*sqrt(a +
b*x**2)/(35*b**3) + 16*a**2*e*x**4*sqrt(a + b*x**2)/(105*b**3) + 80*a**2*f*x**6*sqrt(a + b*x**2)/(693*b**3) -
4*a*c*x**2*sqrt(a + b*x**2)/(15*b**2) - 6*a*d*x**4*sqrt(a + b*x**2)/(35*b**2) - 8*a*e*x**6*sqrt(a + b*x**2)/(6
3*b**2) - 10*a*f*x**8*sqrt(a + b*x**2)/(99*b**2) + c*x**4*sqrt(a + b*x**2)/(5*b) + d*x**6*sqrt(a + b*x**2)/(7*
b) + e*x**8*sqrt(a + b*x**2)/(9*b) + f*x**10*sqrt(a + b*x**2)/(11*b), Ne(b, 0)), ((c*x**6/6 + d*x**8/8 + e*x**
10/10 + f*x**12/12)/sqrt(a), True))

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