3.2.66 \(\int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx\)

Optimal. Leaf size=167 \[ \frac {\left (a+b x^2\right )^{5/2} \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac {a \sqrt {a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {\left (a+b x^2\right )^{7/2} (b e-4 a f)}{7 b^5}+\frac {f \left (a+b x^2\right )^{9/2}}{9 b^5} \]

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Rubi [A]  time = 0.17, antiderivative size = 167, 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} \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac {a \sqrt {a+b x^2} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^5}+\frac {\left (a+b x^2\right )^{5/2} \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {\left (a+b x^2\right )^{7/2} (b e-4 a f)}{7 b^5}+\frac {f \left (a+b x^2\right )^{9/2}}{9 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^3 + d*x^5 + e*x^7 + f*x^9)/Sqrt[a + b*x^2],x]

[Out]

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

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

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Mathematica [A]  time = 0.13, size = 122, normalized size = 0.73 \begin {gather*} \frac {\sqrt {a+b x^2} \left (128 a^4 f-16 a^3 b \left (9 e+4 f x^2\right )+24 a^2 b^2 \left (7 d+3 e x^2+2 f x^4\right )-2 a b^3 \left (105 c+42 d x^2+27 e x^4+20 f x^6\right )+b^4 x^2 \left (105 c+63 d x^2+45 e x^4+35 f x^6\right )\right )}{315 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x^3 + d*x^5 + e*x^7 + f*x^9)/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(128*a^4*f - 16*a^3*b*(9*e + 4*f*x^2) + 24*a^2*b^2*(7*d + 3*e*x^2 + 2*f*x^4) - 2*a*b^3*(105*c
+ 42*d*x^2 + 27*e*x^4 + 20*f*x^6) + b^4*x^2*(105*c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5)

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IntegrateAlgebraic [A]  time = 0.08, size = 148, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a+b x^2} \left (128 a^4 f-144 a^3 b e-64 a^3 b f x^2+168 a^2 b^2 d+72 a^2 b^2 e x^2+48 a^2 b^2 f x^4-210 a b^3 c-84 a b^3 d x^2-54 a b^3 e x^4-40 a b^3 f x^6+105 b^4 c x^2+63 b^4 d x^4+45 b^4 e x^6+35 b^4 f x^8\right )}{315 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*x^3 + d*x^5 + e*x^7 + f*x^9)/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(-210*a*b^3*c + 168*a^2*b^2*d - 144*a^3*b*e + 128*a^4*f + 105*b^4*c*x^2 - 84*a*b^3*d*x^2 + 72
*a^2*b^2*e*x^2 - 64*a^3*b*f*x^2 + 63*b^4*d*x^4 - 54*a*b^3*e*x^4 + 48*a^2*b^2*f*x^4 + 45*b^4*e*x^6 - 40*a*b^3*f
*x^6 + 35*b^4*f*x^8))/(315*b^5)

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fricas [A]  time = 1.11, size = 134, normalized size = 0.80 \begin {gather*} \frac {{\left (35 \, b^{4} f x^{8} + 5 \, {\left (9 \, b^{4} e - 8 \, a b^{3} f\right )} x^{6} - 210 \, a b^{3} c + 168 \, a^{2} b^{2} d - 144 \, a^{3} b e + 128 \, a^{4} f + 3 \, {\left (21 \, b^{4} d - 18 \, a b^{3} e + 16 \, a^{2} b^{2} f\right )} x^{4} + {\left (105 \, b^{4} c - 84 \, a b^{3} d + 72 \, a^{2} b^{2} e - 64 \, a^{3} b f\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^7+d*x^5+c*x^3)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/315*(35*b^4*f*x^8 + 5*(9*b^4*e - 8*a*b^3*f)*x^6 - 210*a*b^3*c + 168*a^2*b^2*d - 144*a^3*b*e + 128*a^4*f + 3*
(21*b^4*d - 18*a*b^3*e + 16*a^2*b^2*f)*x^4 + (105*b^4*c - 84*a*b^3*d + 72*a^2*b^2*e - 64*a^3*b*f)*x^2)*sqrt(b*
x^2 + a)/b^5

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giac [A]  time = 0.50, size = 197, normalized size = 1.18 \begin {gather*} -\frac {{\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \sqrt {b x^{2} + a}}{b^{5}} + \frac {105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} c + 63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} d - 210 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2} d + 35 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} f - 180 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a f + 378 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} f - 420 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} f + 45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b e - 189 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b e + 315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b e}{315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^7+d*x^5+c*x^3)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

-(a*b^3*c - a^2*b^2*d - a^4*f + a^3*b*e)*sqrt(b*x^2 + a)/b^5 + 1/315*(105*(b*x^2 + a)^(3/2)*b^3*c + 63*(b*x^2
+ a)^(5/2)*b^2*d - 210*(b*x^2 + a)^(3/2)*a*b^2*d + 35*(b*x^2 + a)^(9/2)*f - 180*(b*x^2 + a)^(7/2)*a*f + 378*(b
*x^2 + a)^(5/2)*a^2*f - 420*(b*x^2 + a)^(3/2)*a^3*f + 45*(b*x^2 + a)^(7/2)*b*e - 189*(b*x^2 + a)^(5/2)*a*b*e +
315*(b*x^2 + a)^(3/2)*a^2*b*e)/b^5

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maple [A]  time = 0.01, size = 145, normalized size = 0.87 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, \left (35 f \,x^{8} b^{4}-40 a \,b^{3} f \,x^{6}+45 b^{4} e \,x^{6}+48 a^{2} b^{2} f \,x^{4}-54 a \,b^{3} e \,x^{4}+63 b^{4} d \,x^{4}-64 a^{3} b f \,x^{2}+72 a^{2} b^{2} e \,x^{2}-84 a \,b^{3} d \,x^{2}+105 b^{4} c \,x^{2}+128 a^{4} f -144 a^{3} b e +168 a^{2} b^{2} d -210 a \,b^{3} c \right )}{315 b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^7+d*x^5+c*x^3)/(b*x^2+a)^(1/2),x)

[Out]

1/315*(b*x^2+a)^(1/2)*(35*b^4*f*x^8-40*a*b^3*f*x^6+45*b^4*e*x^6+48*a^2*b^2*f*x^4-54*a*b^3*e*x^4+63*b^4*d*x^4-6
4*a^3*b*f*x^2+72*a^2*b^2*e*x^2-84*a*b^3*d*x^2+105*b^4*c*x^2+128*a^4*f-144*a^3*b*e+168*a^2*b^2*d-210*a*b^3*c)/b
^5

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maxima [A]  time = 1.37, size = 263, normalized size = 1.57 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{8}}{9 \, b} + \frac {\sqrt {b x^{2} + a} e x^{6}}{7 \, b} - \frac {8 \, \sqrt {b x^{2} + a} a f x^{6}}{63 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{4}}{5 \, b} - \frac {6 \, \sqrt {b x^{2} + a} a e x^{4}}{35 \, b^{2}} + \frac {16 \, \sqrt {b x^{2} + a} a^{2} f x^{4}}{105 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x^{2}}{3 \, b} - \frac {4 \, \sqrt {b x^{2} + a} a d x^{2}}{15 \, b^{2}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} e x^{2}}{35 \, b^{3}} - \frac {64 \, \sqrt {b x^{2} + a} a^{3} f x^{2}}{315 \, b^{4}} - \frac {2 \, \sqrt {b x^{2} + a} a c}{3 \, b^{2}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} d}{15 \, b^{3}} - \frac {16 \, \sqrt {b x^{2} + a} a^{3} e}{35 \, b^{4}} + \frac {128 \, \sqrt {b x^{2} + a} a^{4} f}{315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^7+d*x^5+c*x^3)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.14, size = 146, normalized size = 0.87 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {128\,f\,a^4-144\,e\,a^3\,b+168\,d\,a^2\,b^2-210\,c\,a\,b^3}{315\,b^5}+\frac {x^4\,\left (48\,f\,a^2\,b^2-54\,e\,a\,b^3+63\,d\,b^4\right )}{315\,b^5}+\frac {f\,x^8}{9\,b}+\frac {x^6\,\left (45\,b^4\,e-40\,a\,b^3\,f\right )}{315\,b^5}+\frac {x^2\,\left (-64\,f\,a^3\,b+72\,e\,a^2\,b^2-84\,d\,a\,b^3+105\,c\,b^4\right )}{315\,b^5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^3 + d*x^5 + e*x^7 + f*x^9)/(a + b*x^2)^(1/2),x)

[Out]

(a + b*x^2)^(1/2)*((128*a^4*f + 168*a^2*b^2*d - 210*a*b^3*c - 144*a^3*b*e)/(315*b^5) + (x^4*(63*b^4*d + 48*a^2
*b^2*f - 54*a*b^3*e))/(315*b^5) + (f*x^8)/(9*b) + (x^6*(45*b^4*e - 40*a*b^3*f))/(315*b^5) + (x^2*(105*b^4*c +
72*a^2*b^2*e - 84*a*b^3*d - 64*a^3*b*f))/(315*b^5))

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sympy [A]  time = 5.41, size = 340, normalized size = 2.04 \begin {gather*} \begin {cases} \frac {128 a^{4} f \sqrt {a + b x^{2}}}{315 b^{5}} - \frac {16 a^{3} e \sqrt {a + b x^{2}}}{35 b^{4}} - \frac {64 a^{3} f x^{2} \sqrt {a + b x^{2}}}{315 b^{4}} + \frac {8 a^{2} d \sqrt {a + b x^{2}}}{15 b^{3}} + \frac {8 a^{2} e x^{2} \sqrt {a + b x^{2}}}{35 b^{3}} + \frac {16 a^{2} f x^{4} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {2 a c \sqrt {a + b x^{2}}}{3 b^{2}} - \frac {4 a d x^{2} \sqrt {a + b x^{2}}}{15 b^{2}} - \frac {6 a e x^{4} \sqrt {a + b x^{2}}}{35 b^{2}} - \frac {8 a f x^{6} \sqrt {a + b x^{2}}}{63 b^{2}} + \frac {c x^{2} \sqrt {a + b x^{2}}}{3 b} + \frac {d x^{4} \sqrt {a + b x^{2}}}{5 b} + \frac {e x^{6} \sqrt {a + b x^{2}}}{7 b} + \frac {f x^{8} \sqrt {a + b x^{2}}}{9 b} & \text {for}\: b \neq 0 \\\frac {\frac {c x^{4}}{4} + \frac {d x^{6}}{6} + \frac {e x^{8}}{8} + \frac {f x^{10}}{10}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**9+e*x**7+d*x**5+c*x**3)/(b*x**2+a)**(1/2),x)

[Out]

Piecewise((128*a**4*f*sqrt(a + b*x**2)/(315*b**5) - 16*a**3*e*sqrt(a + b*x**2)/(35*b**4) - 64*a**3*f*x**2*sqrt
(a + b*x**2)/(315*b**4) + 8*a**2*d*sqrt(a + b*x**2)/(15*b**3) + 8*a**2*e*x**2*sqrt(a + b*x**2)/(35*b**3) + 16*
a**2*f*x**4*sqrt(a + b*x**2)/(105*b**3) - 2*a*c*sqrt(a + b*x**2)/(3*b**2) - 4*a*d*x**2*sqrt(a + b*x**2)/(15*b*
*2) - 6*a*e*x**4*sqrt(a + b*x**2)/(35*b**2) - 8*a*f*x**6*sqrt(a + b*x**2)/(63*b**2) + c*x**2*sqrt(a + b*x**2)/
(3*b) + d*x**4*sqrt(a + b*x**2)/(5*b) + e*x**6*sqrt(a + b*x**2)/(7*b) + f*x**8*sqrt(a + b*x**2)/(9*b), Ne(b, 0
)), ((c*x**4/4 + d*x**6/6 + e*x**8/8 + f*x**10/10)/sqrt(a), True))

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