3.9.84 \(\int e^{\tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [884]

Optimal. Leaf size=105 \[ \frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {5 c^3 \text {ArcSin}(a x)}{16 a} \]

[Out]

5/24*c^3*x*(-a^2*x^2+1)^(3/2)+1/6*c^3*x*(-a^2*x^2+1)^(5/2)-1/7*c^3*(-a^2*x^2+1)^(7/2)/a+5/16*c^3*arcsin(a*x)/a
+5/16*c^3*x*(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6273, 655, 201, 222} \begin {gather*} -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \text {ArcSin}(a x)}{16 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - a^2*c*x^2)^3,x]

[Out]

(5*c^3*x*Sqrt[1 - a^2*x^2])/16 + (5*c^3*x*(1 - a^2*x^2)^(3/2))/24 + (c^3*x*(1 - a^2*x^2)^(5/2))/6 - (c^3*(1 -
a^2*x^2)^(7/2))/(7*a) + (5*c^3*ArcSin[a*x])/(16*a)

Rule 201

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 222

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

Rule 655

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 6273

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a^2*x^2)^(p - n
/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] &&
!IntegerQ[p - n/2]

Rubi steps

\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1+a x) \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+c^3 \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} \left (5 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{8} \left (5 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{16} \left (5 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 91, normalized size = 0.87 \begin {gather*} \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (-48+231 a x+144 a^2 x^2-182 a^3 x^3-144 a^4 x^4+56 a^5 x^5+48 a^6 x^6\right )-210 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{336 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*(c - a^2*c*x^2)^3,x]

[Out]

(c^3*(Sqrt[1 - a^2*x^2]*(-48 + 231*a*x + 144*a^2*x^2 - 182*a^3*x^3 - 144*a^4*x^4 + 56*a^5*x^5 + 48*a^6*x^6) -
210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(336*a)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(87)=174\).
time = 0.07, size = 495, normalized size = 4.71

method	result	size
risch	\(-\frac {\left (48 x^{6} a^{6}+56 x^{5} a^{5}-144 a^{4} x^{4}-182 a^{3} x^{3}+144 a^{2} x^{2}+231 a x -48\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{336 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\)	\(107\)
meijerg	\(-\frac {c^{3} \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{6} a^{6}+48 a^{4} x^{4}+64 a^{2} x^{2}+128\right ) \sqrt {-a^{2} x^{2}+1}}{140}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{3} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{3} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a \sqrt {\pi }}-\frac {c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{3} \arcsin \left (a x \right )}{a}\)	\(419\)
default	\(-c^{3} \left (a^{7} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )+a^{6} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )-3 a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+3 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{a}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )\)	\(495\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-c^3*(a^7*(-1/7*x^6/a^2*(-a^2*x^2+1)^(1/2)+6/7/a^2*(-1/5*x^4/a^2*(-a^2*x^2+1)^(1/2)+4/5/a^2*(-1/3*x^2*(-a^2*x^
2+1)^(1/2)/a^2-2/3*(-a^2*x^2+1)^(1/2)/a^4)))+a^6*(-1/6*x^5/a^2*(-a^2*x^2+1)^(1/2)+5/6/a^2*(-1/4*x^3*(-a^2*x^2+
1)^(1/2)/a^2+3/4/a^2*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2
)))))-3*a^5*(-1/5*x^4/a^2*(-a^2*x^2+1)^(1/2)+4/5/a^2*(-1/3*x^2*(-a^2*x^2+1)^(1/2)/a^2-2/3*(-a^2*x^2+1)^(1/2)/a
^4))-3*a^4*(-1/4*x^3*(-a^2*x^2+1)^(1/2)/a^2+3/4/a^2*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2/a^2/(a^2)^(1/2)*arctan(
(a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))))+3*a^3*(-1/3*x^2*(-a^2*x^2+1)^(1/2)/a^2-2/3*(-a^2*x^2+1)^(1/2)/a^4)+3*a^2*(
-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))+(-a^2*x^2+1)^(1/2)
/a-1/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.46, size = 164, normalized size = 1.56 \begin {gather*} \frac {1}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{3} x^{6} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{5} - \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{4} - \frac {13}{24} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{3} + \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2} + \frac {11}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {5 \, c^{3} \arcsin \left (a x\right )}{16 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{7 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

1/7*sqrt(-a^2*x^2 + 1)*a^5*c^3*x^6 + 1/6*sqrt(-a^2*x^2 + 1)*a^4*c^3*x^5 - 3/7*sqrt(-a^2*x^2 + 1)*a^3*c^3*x^4 -
13/24*sqrt(-a^2*x^2 + 1)*a^2*c^3*x^3 + 3/7*sqrt(-a^2*x^2 + 1)*a*c^3*x^2 + 11/16*sqrt(-a^2*x^2 + 1)*c^3*x + 5/
16*c^3*arcsin(a*x)/a - 1/7*sqrt(-a^2*x^2 + 1)*c^3/a

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Fricas [A]
time = 0.40, size = 115, normalized size = 1.10 \begin {gather*} -\frac {210 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (48 \, a^{6} c^{3} x^{6} + 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} - 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} + 231 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/336*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (48*a^6*c^3*x^6 + 56*a^5*c^3*x^5 - 144*a^4*c^3*x^4 -
182*a^3*c^3*x^3 + 144*a^2*c^3*x^2 + 231*a*c^3*x - 48*c^3)*sqrt(-a^2*x^2 + 1))/a

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (90) = 180\).
time = 9.18, size = 377, normalized size = 3.59 \begin {gather*} \begin {cases} \frac {- c^{3} \sqrt {- a^{2} x^{2} + 1} - 3 c^{3} \left (\begin {cases} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 3 c^{3} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 3 c^{3} \left (\begin {cases} \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 3 c^{3} \left (\begin {cases} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \frac {2 \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{6} + \frac {3 a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{16} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {7}{2}}}{7} - \frac {3 \left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{3} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{3} x & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**3,x)

[Out]

Piecewise(((-c**3*sqrt(-a**2*x**2 + 1) - 3*c**3*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -
1) & (a*x < 1))) - 3*c**3*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))
) + 3*c**3*Piecewise((a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 - a*x*sqrt(-a**2*x**2 + 1)/2 + 3*asin(a*x)
/8, (a*x > -1) & (a*x < 1))) + 3*c**3*Piecewise((-(-a**2*x**2 + 1)**(5/2)/5 + 2*(-a**2*x**2 + 1)**(3/2)/3 - sq
rt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) - c**3*Piecewise((a**3*x**3*(-a**2*x**2 + 1)**(3/2)/6 + 3*a*x*(-2
*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/16 - a*x*sqrt(-a**2*x**2 + 1)/2 + 5*asin(a*x)/16, (a*x > -1) & (a*x < 1))
) - c**3*Piecewise(((-a**2*x**2 + 1)**(7/2)/7 - 3*(-a**2*x**2 + 1)**(5/2)/5 + (-a**2*x**2 + 1)**(3/2) - sqrt(-
a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c**3*asin(a*x))/a, Ne(a, 0)), (c**3*x, True))

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Giac [A]
time = 0.45, size = 103, normalized size = 0.98 \begin {gather*} \frac {5 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, {\left | a \right |}} - \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {48 \, c^{3}}{a} - {\left (231 \, c^{3} + 2 \, {\left (72 \, a c^{3} - {\left (91 \, a^{2} c^{3} + 4 \, {\left (18 \, a^{3} c^{3} - {\left (6 \, a^{5} c^{3} x + 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

5/16*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/336*sqrt(-a^2*x^2 + 1)*(48*c^3/a - (231*c^3 + 2*(72*a*c^3 - (91*a^2*c^3
+ 4*(18*a^3*c^3 - (6*a^5*c^3*x + 7*a^4*c^3)*x)*x)*x)*x)*x)

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Mupad [B]
time = 0.90, size = 100, normalized size = 0.95 \begin {gather*} \frac {5\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {5\,c^3\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{24}+\frac {c^3\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{6}-\frac {c^3\,{\left (1-a^2\,x^2\right )}^{7/2}}{7\,a}-\frac {5\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{16\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a^2*c*x^2)^3*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)

[Out]

(5*c^3*x*(1 - a^2*x^2)^(1/2))/16 + (5*c^3*x*(1 - a^2*x^2)^(3/2))/24 + (c^3*x*(1 - a^2*x^2)^(5/2))/6 - (c^3*(1
- a^2*x^2)^(7/2))/(7*a) - (5*c^3*asinh(x*(-a^2)^(1/2))*(-a^2)^(1/2))/(16*a^2)

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