3.7.29 \(\int (a+b x)^3 (a^2-b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=164 \[ -\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {9 a^7 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b}+\frac {9}{16} a^5 x \sqrt {a^2-b^2 x^2}+\frac {3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2} \]

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Rubi [A]  time = 0.06, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \begin {gather*} \frac {9}{16} a^5 x \sqrt {a^2-b^2 x^2}+\frac {3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {9 a^7 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(a^2 - b^2*x^2)^(3/2),x]

[Out]

(9*a^5*x*Sqrt[a^2 - b^2*x^2])/16 + (3*a^3*x*(a^2 - b^2*x^2)^(3/2))/8 - (3*a^2*(a^2 - b^2*x^2)^(5/2))/(10*b) -
(3*a*(a + b*x)*(a^2 - b^2*x^2)^(5/2))/(14*b) - ((a + b*x)^2*(a^2 - b^2*x^2)^(5/2))/(7*b) + (9*a^7*ArcTan[(b*x)
/Sqrt[a^2 - b^2*x^2]])/(16*b)

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rubi steps

\begin {align*} \int (a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2} \, dx &=-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {1}{7} (9 a) \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {1}{2} \left (3 a^2\right ) \int (a+b x) \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=-\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {1}{2} \left (3 a^3\right ) \int \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=\frac {3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {1}{8} \left (9 a^5\right ) \int \sqrt {a^2-b^2 x^2} \, dx\\ &=\frac {9}{16} a^5 x \sqrt {a^2-b^2 x^2}+\frac {3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {1}{16} \left (9 a^7\right ) \int \frac {1}{\sqrt {a^2-b^2 x^2}} \, dx\\ &=\frac {9}{16} a^5 x \sqrt {a^2-b^2 x^2}+\frac {3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {1}{16} \left (9 a^7\right ) \operatorname {Subst}\left (\int \frac {1}{1+b^2 x^2} \, dx,x,\frac {x}{\sqrt {a^2-b^2 x^2}}\right )\\ &=\frac {9}{16} a^5 x \sqrt {a^2-b^2 x^2}+\frac {3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac {3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac {3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac {9 a^7 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 134, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (315 a^6 \sin ^{-1}\left (\frac {b x}{a}\right )+\sqrt {1-\frac {b^2 x^2}{a^2}} \left (-368 a^6+245 a^5 b x+656 a^4 b^2 x^2+350 a^3 b^3 x^3-208 a^2 b^4 x^4-280 a b^5 x^5-80 b^6 x^6\right )\right )}{560 b \sqrt {1-\frac {b^2 x^2}{a^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(a^2 - b^2*x^2)^(3/2),x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-368*a^6 + 245*a^5*b*x + 656*a^4*b^2*x^2 + 350*a^3*b^3*x^3 - 20
8*a^2*b^4*x^4 - 280*a*b^5*x^5 - 80*b^6*x^6) + 315*a^6*ArcSin[(b*x)/a]))/(560*b*Sqrt[1 - (b^2*x^2)/a^2])

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IntegrateAlgebraic [A]  time = 0.43, size = 136, normalized size = 0.83 \begin {gather*} \frac {9 a^7 \sqrt {-b^2} \log \left (\sqrt {a^2-b^2 x^2}-\sqrt {-b^2} x\right )}{16 b^2}+\frac {\sqrt {a^2-b^2 x^2} \left (-368 a^6+245 a^5 b x+656 a^4 b^2 x^2+350 a^3 b^3 x^3-208 a^2 b^4 x^4-280 a b^5 x^5-80 b^6 x^6\right )}{560 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^3*(a^2 - b^2*x^2)^(3/2),x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-368*a^6 + 245*a^5*b*x + 656*a^4*b^2*x^2 + 350*a^3*b^3*x^3 - 208*a^2*b^4*x^4 - 280*a*b^5
*x^5 - 80*b^6*x^6))/(560*b) + (9*a^7*Sqrt[-b^2]*Log[-(Sqrt[-b^2]*x) + Sqrt[a^2 - b^2*x^2]])/(16*b^2)

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fricas [A]  time = 0.42, size = 116, normalized size = 0.71 \begin {gather*} -\frac {630 \, a^{7} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + {\left (80 \, b^{6} x^{6} + 280 \, a b^{5} x^{5} + 208 \, a^{2} b^{4} x^{4} - 350 \, a^{3} b^{3} x^{3} - 656 \, a^{4} b^{2} x^{2} - 245 \, a^{5} b x + 368 \, a^{6}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{560 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(-b^2*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/560*(630*a^7*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) + (80*b^6*x^6 + 280*a*b^5*x^5 + 208*a^2*b^4*x^4 - 35
0*a^3*b^3*x^3 - 656*a^4*b^2*x^2 - 245*a^5*b*x + 368*a^6)*sqrt(-b^2*x^2 + a^2))/b

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giac [A]  time = 0.32, size = 104, normalized size = 0.63 \begin {gather*} \frac {9 \, a^{7} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (b)}{16 \, {\left | b \right |}} - \frac {1}{560} \, {\left (\frac {368 \, a^{6}}{b} - {\left (245 \, a^{5} + 2 \, {\left (328 \, a^{4} b + {\left (175 \, a^{3} b^{2} - 4 \, {\left (26 \, a^{2} b^{3} + 5 \, {\left (2 \, b^{5} x + 7 \, a b^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-b^{2} x^{2} + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(-b^2*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

9/16*a^7*arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) - 1/560*(368*a^6/b - (245*a^5 + 2*(328*a^4*b + (175*a^3*b^2 - 4*(2
6*a^2*b^3 + 5*(2*b^5*x + 7*a*b^4)*x)*x)*x)*x)*x)*sqrt(-b^2*x^2 + a^2)

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maple [A]  time = 0.08, size = 134, normalized size = 0.82 \begin {gather*} \frac {9 a^{7} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{16 \sqrt {b^{2}}}+\frac {9 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{5} x}{16}+\frac {3 \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}} a^{3} x}{8}-\frac {\left (-b^{2} x^{2}+a^{2}\right )^{\frac {5}{2}} b \,x^{2}}{7}-\frac {\left (-b^{2} x^{2}+a^{2}\right )^{\frac {5}{2}} a x}{2}-\frac {23 \left (-b^{2} x^{2}+a^{2}\right )^{\frac {5}{2}} a^{2}}{35 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(-b^2*x^2+a^2)^(3/2),x)

[Out]

-1/7*b*x^2*(-b^2*x^2+a^2)^(5/2)-23/35*a^2*(-b^2*x^2+a^2)^(5/2)/b-1/2*a*x*(-b^2*x^2+a^2)^(5/2)+3/8*a^3*x*(-b^2*
x^2+a^2)^(3/2)+9/16*a^5*x*(-b^2*x^2+a^2)^(1/2)+9/16*a^7/(b^2)^(1/2)*arctan((b^2)^(1/2)/(-b^2*x^2+a^2)^(1/2)*x)

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maxima [A]  time = 2.97, size = 116, normalized size = 0.71 \begin {gather*} \frac {9 \, a^{7} \arcsin \left (\frac {b x}{a}\right )}{16 \, b} + \frac {9}{16} \, \sqrt {-b^{2} x^{2} + a^{2}} a^{5} x + \frac {3}{8} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a^{3} x - \frac {1}{7} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {5}{2}} b x^{2} - \frac {1}{2} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {5}{2}} a x - \frac {23 \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {5}{2}} a^{2}}{35 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(-b^2*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

9/16*a^7*arcsin(b*x/a)/b + 9/16*sqrt(-b^2*x^2 + a^2)*a^5*x + 3/8*(-b^2*x^2 + a^2)^(3/2)*a^3*x - 1/7*(-b^2*x^2
+ a^2)^(5/2)*b*x^2 - 1/2*(-b^2*x^2 + a^2)^(5/2)*a*x - 23/35*(-b^2*x^2 + a^2)^(5/2)*a^2/b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a^2-b^2\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 - b^2*x^2)^(3/2)*(a + b*x)^3,x)

[Out]

int((a^2 - b^2*x^2)^(3/2)*(a + b*x)^3, x)

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sympy [C]  time = 16.61, size = 816, normalized size = 4.98 \begin {gather*} a^{5} \left (\begin {cases} - \frac {i a^{2} \operatorname {acosh}{\left (\frac {b x}{a} \right )}}{2 b} - \frac {i a x}{2 \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} + \frac {i b^{2} x^{3}}{2 a \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} & \text {for}\: \left |{\frac {b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac {a^{2} \operatorname {asin}{\left (\frac {b x}{a} \right )}}{2 b} + \frac {a x \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}}{2} & \text {otherwise} \end {cases}\right ) + 3 a^{4} b \left (\begin {cases} \frac {x^{2} \sqrt {a^{2}}}{2} & \text {for}\: b^{2} = 0 \\- \frac {\left (a^{2} - b^{2} x^{2}\right )^{\frac {3}{2}}}{3 b^{2}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} b^{2} \left (\begin {cases} - \frac {i a^{4} \operatorname {acosh}{\left (\frac {b x}{a} \right )}}{8 b^{3}} + \frac {i a^{3} x}{8 b^{2} \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} - \frac {3 i a x^{3}}{8 \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} + \frac {i b^{2} x^{5}}{4 a \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} & \text {for}\: \left |{\frac {b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac {a^{4} \operatorname {asin}{\left (\frac {b x}{a} \right )}}{8 b^{3}} - \frac {a^{3} x}{8 b^{2} \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} + \frac {3 a x^{3}}{8 \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} - \frac {b^{2} x^{5}}{4 a \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} & \text {otherwise} \end {cases}\right ) - 2 a^{2} b^{3} \left (\begin {cases} - \frac {2 a^{4} \sqrt {a^{2} - b^{2} x^{2}}}{15 b^{4}} - \frac {a^{2} x^{2} \sqrt {a^{2} - b^{2} x^{2}}}{15 b^{2}} + \frac {x^{4} \sqrt {a^{2} - b^{2} x^{2}}}{5} & \text {for}\: b \neq 0 \\\frac {x^{4} \sqrt {a^{2}}}{4} & \text {otherwise} \end {cases}\right ) - 3 a b^{4} \left (\begin {cases} - \frac {i a^{6} \operatorname {acosh}{\left (\frac {b x}{a} \right )}}{16 b^{5}} + \frac {i a^{5} x}{16 b^{4} \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} - \frac {i a^{3} x^{3}}{48 b^{2} \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} - \frac {5 i a x^{5}}{24 \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} + \frac {i b^{2} x^{7}}{6 a \sqrt {-1 + \frac {b^{2} x^{2}}{a^{2}}}} & \text {for}\: \left |{\frac {b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac {a^{6} \operatorname {asin}{\left (\frac {b x}{a} \right )}}{16 b^{5}} - \frac {a^{5} x}{16 b^{4} \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} + \frac {a^{3} x^{3}}{48 b^{2} \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} + \frac {5 a x^{5}}{24 \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} - \frac {b^{2} x^{7}}{6 a \sqrt {1 - \frac {b^{2} x^{2}}{a^{2}}}} & \text {otherwise} \end {cases}\right ) - b^{5} \left (\begin {cases} - \frac {8 a^{6} \sqrt {a^{2} - b^{2} x^{2}}}{105 b^{6}} - \frac {4 a^{4} x^{2} \sqrt {a^{2} - b^{2} x^{2}}}{105 b^{4}} - \frac {a^{2} x^{4} \sqrt {a^{2} - b^{2} x^{2}}}{35 b^{2}} + \frac {x^{6} \sqrt {a^{2} - b^{2} x^{2}}}{7} & \text {for}\: b \neq 0 \\\frac {x^{6} \sqrt {a^{2}}}{6} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(-b**2*x**2+a**2)**(3/2),x)

[Out]

a**5*Piecewise((-I*a**2*acosh(b*x/a)/(2*b) - I*a*x/(2*sqrt(-1 + b**2*x**2/a**2)) + I*b**2*x**3/(2*a*sqrt(-1 +
b**2*x**2/a**2)), Abs(b**2*x**2/a**2) > 1), (a**2*asin(b*x/a)/(2*b) + a*x*sqrt(1 - b**2*x**2/a**2)/2, True)) +
3*a**4*b*Piecewise((x**2*sqrt(a**2)/2, Eq(b**2, 0)), (-(a**2 - b**2*x**2)**(3/2)/(3*b**2), True)) + 2*a**3*b*
*2*Piecewise((-I*a**4*acosh(b*x/a)/(8*b**3) + I*a**3*x/(8*b**2*sqrt(-1 + b**2*x**2/a**2)) - 3*I*a*x**3/(8*sqrt
(-1 + b**2*x**2/a**2)) + I*b**2*x**5/(4*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2*x**2/a**2) > 1), (a**4*asin(b*x
/a)/(8*b**3) - a**3*x/(8*b**2*sqrt(1 - b**2*x**2/a**2)) + 3*a*x**3/(8*sqrt(1 - b**2*x**2/a**2)) - b**2*x**5/(4
*a*sqrt(1 - b**2*x**2/a**2)), True)) - 2*a**2*b**3*Piecewise((-2*a**4*sqrt(a**2 - b**2*x**2)/(15*b**4) - a**2*
x**2*sqrt(a**2 - b**2*x**2)/(15*b**2) + x**4*sqrt(a**2 - b**2*x**2)/5, Ne(b, 0)), (x**4*sqrt(a**2)/4, True)) -
3*a*b**4*Piecewise((-I*a**6*acosh(b*x/a)/(16*b**5) + I*a**5*x/(16*b**4*sqrt(-1 + b**2*x**2/a**2)) - I*a**3*x*
*3/(48*b**2*sqrt(-1 + b**2*x**2/a**2)) - 5*I*a*x**5/(24*sqrt(-1 + b**2*x**2/a**2)) + I*b**2*x**7/(6*a*sqrt(-1
+ b**2*x**2/a**2)), Abs(b**2*x**2/a**2) > 1), (a**6*asin(b*x/a)/(16*b**5) - a**5*x/(16*b**4*sqrt(1 - b**2*x**2
/a**2)) + a**3*x**3/(48*b**2*sqrt(1 - b**2*x**2/a**2)) + 5*a*x**5/(24*sqrt(1 - b**2*x**2/a**2)) - b**2*x**7/(6
*a*sqrt(1 - b**2*x**2/a**2)), True)) - b**5*Piecewise((-8*a**6*sqrt(a**2 - b**2*x**2)/(105*b**6) - 4*a**4*x**2
*sqrt(a**2 - b**2*x**2)/(105*b**4) - a**2*x**4*sqrt(a**2 - b**2*x**2)/(35*b**2) + x**6*sqrt(a**2 - b**2*x**2)/
7, Ne(b, 0)), (x**6*sqrt(a**2)/6, True))

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