3.425 \(\int x^6 (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=202 \[ -\frac {45 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{32768 b^{7/2}}+\frac {45 a^7 x \sqrt {a+b x^2}}{32768 b^3}-\frac {15 a^6 x^3 \sqrt {a+b x^2}}{16384 b^2}+\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2} \]

[Out]

3/256*a^3*x^7*(b*x^2+a)^(3/2)+3/128*a^2*x^7*(b*x^2+a)^(5/2)+9/224*a*x^7*(b*x^2+a)^(7/2)+1/16*x^7*(b*x^2+a)^(9/
2)-45/32768*a^8*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(7/2)+45/32768*a^7*x*(b*x^2+a)^(1/2)/b^3-15/16384*a^6*x^3
*(b*x^2+a)^(1/2)/b^2+3/4096*a^5*x^5*(b*x^2+a)^(1/2)/b+9/2048*a^4*x^7*(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ \frac {45 a^7 x \sqrt {a+b x^2}}{32768 b^3}-\frac {15 a^6 x^3 \sqrt {a+b x^2}}{16384 b^2}-\frac {45 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{32768 b^{7/2}}+\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x^2)^(9/2),x]

[Out]

(45*a^7*x*Sqrt[a + b*x^2])/(32768*b^3) - (15*a^6*x^3*Sqrt[a + b*x^2])/(16384*b^2) + (3*a^5*x^5*Sqrt[a + b*x^2]
)/(4096*b) + (9*a^4*x^7*Sqrt[a + b*x^2])/2048 + (3*a^3*x^7*(a + b*x^2)^(3/2))/256 + (3*a^2*x^7*(a + b*x^2)^(5/
2))/128 + (9*a*x^7*(a + b*x^2)^(7/2))/224 + (x^7*(a + b*x^2)^(9/2))/16 - (45*a^8*ArcTanh[(Sqrt[b]*x)/Sqrt[a +
b*x^2]])/(32768*b^(7/2))

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 279

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

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]

Rubi steps

\begin {align*} \int x^6 \left (a+b x^2\right )^{9/2} \, dx &=\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac {1}{16} (9 a) \int x^6 \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac {1}{32} \left (9 a^2\right ) \int x^6 \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac {1}{128} \left (15 a^3\right ) \int x^6 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac {1}{256} \left (9 a^4\right ) \int x^6 \sqrt {a+b x^2} \, dx\\ &=\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac {\left (9 a^5\right ) \int \frac {x^6}{\sqrt {a+b x^2}} \, dx}{2048}\\ &=\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac {\left (15 a^6\right ) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{4096 b}\\ &=-\frac {15 a^6 x^3 \sqrt {a+b x^2}}{16384 b^2}+\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac {\left (45 a^7\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{16384 b^2}\\ &=\frac {45 a^7 x \sqrt {a+b x^2}}{32768 b^3}-\frac {15 a^6 x^3 \sqrt {a+b x^2}}{16384 b^2}+\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac {\left (45 a^8\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{32768 b^3}\\ &=\frac {45 a^7 x \sqrt {a+b x^2}}{32768 b^3}-\frac {15 a^6 x^3 \sqrt {a+b x^2}}{16384 b^2}+\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac {\left (45 a^8\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{32768 b^3}\\ &=\frac {45 a^7 x \sqrt {a+b x^2}}{32768 b^3}-\frac {15 a^6 x^3 \sqrt {a+b x^2}}{16384 b^2}+\frac {3 a^5 x^5 \sqrt {a+b x^2}}{4096 b}+\frac {9 a^4 x^7 \sqrt {a+b x^2}}{2048}+\frac {3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac {3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac {9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac {1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac {45 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{32768 b^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 138, normalized size = 0.68 \[ \frac {\sqrt {a+b x^2} \left (\sqrt {b} x \left (315 a^7-210 a^6 b x^2+168 a^5 b^2 x^4+32624 a^4 b^3 x^6+98432 a^3 b^4 x^8+119040 a^2 b^5 x^{10}+66560 a b^6 x^{12}+14336 b^7 x^{14}\right )-\frac {315 a^{15/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{229376 b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(315*a^7 - 210*a^6*b*x^2 + 168*a^5*b^2*x^4 + 32624*a^4*b^3*x^6 + 98432*a^3*b^4*x^8
+ 119040*a^2*b^5*x^10 + 66560*a*b^6*x^12 + 14336*b^7*x^14) - (315*a^(15/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt
[1 + (b*x^2)/a]))/(229376*b^(7/2))

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fricas [A]  time = 1.24, size = 255, normalized size = 1.26 \[ \left [\frac {315 \, a^{8} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (14336 \, b^{8} x^{15} + 66560 \, a b^{7} x^{13} + 119040 \, a^{2} b^{6} x^{11} + 98432 \, a^{3} b^{5} x^{9} + 32624 \, a^{4} b^{4} x^{7} + 168 \, a^{5} b^{3} x^{5} - 210 \, a^{6} b^{2} x^{3} + 315 \, a^{7} b x\right )} \sqrt {b x^{2} + a}}{458752 \, b^{4}}, \frac {315 \, a^{8} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (14336 \, b^{8} x^{15} + 66560 \, a b^{7} x^{13} + 119040 \, a^{2} b^{6} x^{11} + 98432 \, a^{3} b^{5} x^{9} + 32624 \, a^{4} b^{4} x^{7} + 168 \, a^{5} b^{3} x^{5} - 210 \, a^{6} b^{2} x^{3} + 315 \, a^{7} b x\right )} \sqrt {b x^{2} + a}}{229376 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

[1/458752*(315*a^8*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(14336*b^8*x^15 + 66560*a*b^7*x
^13 + 119040*a^2*b^6*x^11 + 98432*a^3*b^5*x^9 + 32624*a^4*b^4*x^7 + 168*a^5*b^3*x^5 - 210*a^6*b^2*x^3 + 315*a^
7*b*x)*sqrt(b*x^2 + a))/b^4, 1/229376*(315*a^8*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (14336*b^8*x^15 +
66560*a*b^7*x^13 + 119040*a^2*b^6*x^11 + 98432*a^3*b^5*x^9 + 32624*a^4*b^4*x^7 + 168*a^5*b^3*x^5 - 210*a^6*b^
2*x^3 + 315*a^7*b*x)*sqrt(b*x^2 + a))/b^4]

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giac [A]  time = 1.21, size = 133, normalized size = 0.66 \[ \frac {45 \, a^{8} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{32768 \, b^{\frac {7}{2}}} + \frac {1}{229376} \, {\left (\frac {315 \, a^{7}}{b^{3}} - 2 \, {\left (\frac {105 \, a^{6}}{b^{2}} - 4 \, {\left (\frac {21 \, a^{5}}{b} + 2 \, {\left (2039 \, a^{4} + 8 \, {\left (769 \, a^{3} b + 2 \, {\left (465 \, a^{2} b^{2} + 4 \, {\left (14 \, b^{4} x^{2} + 65 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

45/32768*a^8*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2) + 1/229376*(315*a^7/b^3 - 2*(105*a^6/b^2 - 4*(21*a
^5/b + 2*(2039*a^4 + 8*(769*a^3*b + 2*(465*a^2*b^2 + 4*(14*b^4*x^2 + 65*a*b^3)*x^2)*x^2)*x^2)*x^2)*x^2)*x^2)*s
qrt(b*x^2 + a)*x

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maple [A]  time = 0.01, size = 169, normalized size = 0.84 \[ -\frac {45 a^{8} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{32768 b^{\frac {7}{2}}}-\frac {45 \sqrt {b \,x^{2}+a}\, a^{7} x}{32768 b^{3}}-\frac {15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{6} x}{16384 b^{3}}+\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} x^{5}}{16 b}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{5} x}{4096 b^{3}}-\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4} x}{14336 b^{3}}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {11}{2}} a \,x^{3}}{224 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}} a^{3} x}{1792 b^{3}}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {11}{2}} a^{2} x}{896 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^2+a)^(9/2),x)

[Out]

1/16*x^5*(b*x^2+a)^(11/2)/b-5/224*a/b^2*x^3*(b*x^2+a)^(11/2)+5/896*a^2/b^3*x*(b*x^2+a)^(11/2)-1/1792*a^3/b^3*x
*(b*x^2+a)^(9/2)-9/14336*a^4/b^3*x*(b*x^2+a)^(7/2)-3/4096*a^5/b^3*x*(b*x^2+a)^(5/2)-15/16384*a^6/b^3*x*(b*x^2+
a)^(3/2)-45/32768*a^7*x*(b*x^2+a)^(1/2)/b^3-45/32768*a^8/b^(7/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.46, size = 161, normalized size = 0.80 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{5}}{16 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{3}}{224 \, b^{2}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2} x}{896 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {9}{2}} a^{3} x}{1792 \, b^{3}} - \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x}{14336 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5} x}{4096 \, b^{3}} - \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6} x}{16384 \, b^{3}} - \frac {45 \, \sqrt {b x^{2} + a} a^{7} x}{32768 \, b^{3}} - \frac {45 \, a^{8} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{32768 \, b^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

1/16*(b*x^2 + a)^(11/2)*x^5/b - 5/224*(b*x^2 + a)^(11/2)*a*x^3/b^2 + 5/896*(b*x^2 + a)^(11/2)*a^2*x/b^3 - 1/17
92*(b*x^2 + a)^(9/2)*a^3*x/b^3 - 9/14336*(b*x^2 + a)^(7/2)*a^4*x/b^3 - 3/4096*(b*x^2 + a)^(5/2)*a^5*x/b^3 - 15
/16384*(b*x^2 + a)^(3/2)*a^6*x/b^3 - 45/32768*sqrt(b*x^2 + a)*a^7*x/b^3 - 45/32768*a^8*arcsinh(b*x/sqrt(a*b))/
b^(7/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^6\,{\left (b\,x^2+a\right )}^{9/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*x^2)^(9/2),x)

[Out]

int(x^6*(a + b*x^2)^(9/2), x)

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sympy [A]  time = 30.60, size = 258, normalized size = 1.28 \[ \frac {45 a^{\frac {15}{2}} x}{32768 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{\frac {13}{2}} x^{3}}{32768 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {11}{2}} x^{5}}{16384 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {4099 a^{\frac {9}{2}} x^{7}}{28672 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8191 a^{\frac {7}{2}} b x^{9}}{14336 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {1699 a^{\frac {5}{2}} b^{2} x^{11}}{1792 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {725 a^{\frac {3}{2}} b^{3} x^{13}}{896 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {79 \sqrt {a} b^{4} x^{15}}{224 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {45 a^{8} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{32768 b^{\frac {7}{2}}} + \frac {b^{5} x^{17}}{16 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**2+a)**(9/2),x)

[Out]

45*a**(15/2)*x/(32768*b**3*sqrt(1 + b*x**2/a)) + 15*a**(13/2)*x**3/(32768*b**2*sqrt(1 + b*x**2/a)) - 3*a**(11/
2)*x**5/(16384*b*sqrt(1 + b*x**2/a)) + 4099*a**(9/2)*x**7/(28672*sqrt(1 + b*x**2/a)) + 8191*a**(7/2)*b*x**9/(1
4336*sqrt(1 + b*x**2/a)) + 1699*a**(5/2)*b**2*x**11/(1792*sqrt(1 + b*x**2/a)) + 725*a**(3/2)*b**3*x**13/(896*s
qrt(1 + b*x**2/a)) + 79*sqrt(a)*b**4*x**15/(224*sqrt(1 + b*x**2/a)) - 45*a**8*asinh(sqrt(b)*x/sqrt(a))/(32768*
b**(7/2)) + b**5*x**17/(16*sqrt(a)*sqrt(1 + b*x**2/a))

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