3.1099 \(\int e^{2 \tanh ^{-1}(a x)} x^2 (c-a^2 c x^2)^{5/2} \, dx\)

Optimal. Leaf size=162 \[ \frac {11 c^2 x \sqrt {c-a^2 c x^2}}{128 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}+\frac {11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}+\frac {11 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a^3}-\frac {(385 a x+192) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3} \]

[Out]

11/192*c*x*(-a^2*c*x^2+c)^(3/2)/a^2-2/7*x^2*(-a^2*c*x^2+c)^(5/2)/a-1/8*x^3*(-a^2*c*x^2+c)^(5/2)-1/1680*(385*a*
x+192)*(-a^2*c*x^2+c)^(5/2)/a^3+11/128*c^(5/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))/a^3+11/128*c^2*x*(-a^2
*c*x^2+c)^(1/2)/a^2

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Rubi [A]  time = 0.33, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1809, 833, 780, 195, 217, 203} \[ \frac {11 c^2 x \sqrt {c-a^2 c x^2}}{128 a^2}+\frac {11 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a^3}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}+\frac {11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac {(385 a x+192) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^(5/2),x]

[Out]

(11*c^2*x*Sqrt[c - a^2*c*x^2])/(128*a^2) + (11*c*x*(c - a^2*c*x^2)^(3/2))/(192*a^2) - (2*x^2*(c - a^2*c*x^2)^(
5/2))/(7*a) - (x^3*(c - a^2*c*x^2)^(5/2))/8 - ((192 + 385*a*x)*(c - a^2*c*x^2)^(5/2))/(1680*a^3) + (11*c^(5/2)
*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a^3)

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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 833

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6151

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

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x^2 (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {\int x^2 \left (-11 a^2 c-16 a^3 c x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{8 a^2}\\ &=-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}+\frac {\int x \left (32 a^3 c^2+77 a^4 c^2 x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{56 a^4 c}\\ &=-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac {(11 c) \int \left (c-a^2 c x^2\right )^{3/2} \, dx}{48 a^2}\\ &=\frac {11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac {\left (11 c^2\right ) \int \sqrt {c-a^2 c x^2} \, dx}{64 a^2}\\ &=\frac {11 c^2 x \sqrt {c-a^2 c x^2}}{128 a^2}+\frac {11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac {\left (11 c^3\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{128 a^2}\\ &=\frac {11 c^2 x \sqrt {c-a^2 c x^2}}{128 a^2}+\frac {11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac {\left (11 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{128 a^2}\\ &=\frac {11 c^2 x \sqrt {c-a^2 c x^2}}{128 a^2}+\frac {11 c x \left (c-a^2 c x^2\right )^{3/2}}{192 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a}-\frac {1}{8} x^3 \left (c-a^2 c x^2\right )^{5/2}-\frac {(192+385 a x) \left (c-a^2 c x^2\right )^{5/2}}{1680 a^3}+\frac {11 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 123, normalized size = 0.76 \[ -\frac {c^2 \left (1155 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )+\left (1680 a^7 x^7+3840 a^6 x^6-280 a^5 x^5-6144 a^4 x^4-3710 a^3 x^3+768 a^2 x^2+1155 a x+1536\right ) \sqrt {c-a^2 c x^2}\right )}{13440 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^(5/2),x]

[Out]

-1/13440*(c^2*(Sqrt[c - a^2*c*x^2]*(1536 + 1155*a*x + 768*a^2*x^2 - 3710*a^3*x^3 - 6144*a^4*x^4 - 280*a^5*x^5
+ 3840*a^6*x^6 + 1680*a^7*x^7) + 1155*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))]))/a^3

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fricas [A]  time = 0.88, size = 285, normalized size = 1.76 \[ \left [\frac {1155 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (1680 \, a^{7} c^{2} x^{7} + 3840 \, a^{6} c^{2} x^{6} - 280 \, a^{5} c^{2} x^{5} - 6144 \, a^{4} c^{2} x^{4} - 3710 \, a^{3} c^{2} x^{3} + 768 \, a^{2} c^{2} x^{2} + 1155 \, a c^{2} x + 1536 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{26880 \, a^{3}}, -\frac {1155 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (1680 \, a^{7} c^{2} x^{7} + 3840 \, a^{6} c^{2} x^{6} - 280 \, a^{5} c^{2} x^{5} - 6144 \, a^{4} c^{2} x^{4} - 3710 \, a^{3} c^{2} x^{3} + 768 \, a^{2} c^{2} x^{2} + 1155 \, a c^{2} x + 1536 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{13440 \, a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

[1/26880*(1155*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(1680*a^7*c^2*x^7 +
3840*a^6*c^2*x^6 - 280*a^5*c^2*x^5 - 6144*a^4*c^2*x^4 - 3710*a^3*c^2*x^3 + 768*a^2*c^2*x^2 + 1155*a*c^2*x + 1
536*c^2)*sqrt(-a^2*c*x^2 + c))/a^3, -1/13440*(1155*c^(5/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2
- c)) + (1680*a^7*c^2*x^7 + 3840*a^6*c^2*x^6 - 280*a^5*c^2*x^5 - 6144*a^4*c^2*x^4 - 3710*a^3*c^2*x^3 + 768*a^2
*c^2*x^2 + 1155*a*c^2*x + 1536*c^2)*sqrt(-a^2*c*x^2 + c))/a^3]

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giac [A]  time = 0.28, size = 143, normalized size = 0.88 \[ \frac {1}{13440} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (1855 \, c^{2} + 4 \, {\left (768 \, a c^{2} + 5 \, {\left (7 \, a^{2} c^{2} - 6 \, {\left (7 \, a^{4} c^{2} x + 16 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {384 \, c^{2}}{a}\right )} x - \frac {1155 \, c^{2}}{a^{2}}\right )} x - \frac {1536 \, c^{2}}{a^{3}}\right )} - \frac {11 \, c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{128 \, a^{2} \sqrt {-c} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/13440*sqrt(-a^2*c*x^2 + c)*((2*((1855*c^2 + 4*(768*a*c^2 + 5*(7*a^2*c^2 - 6*(7*a^4*c^2*x + 16*a^3*c^2)*x)*x)
*x)*x - 384*c^2/a)*x - 1155*c^2/a^2)*x - 1536*c^2/a^3) - 11/128*c^3*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2
+ c)))/(a^2*sqrt(-c)*abs(a))

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maple [B]  time = 0.04, size = 306, normalized size = 1.89 \[ \frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8 a^{2} c}-\frac {17 x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{48 a^{2}}-\frac {85 c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{192 a^{2}}-\frac {85 c^{2} x \sqrt {-a^{2} c \,x^{2}+c}}{128 a^{2}}-\frac {85 c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{128 a^{2} \sqrt {a^{2} c}}+\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{7 a^{3} c}-\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{5 a^{3}}+\frac {c \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{2 a^{2}}+\frac {3 c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{4 a^{2}}+\frac {3 c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{4 a^{2} \sqrt {a^{2} c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(5/2),x)

[Out]

1/8*x*(-a^2*c*x^2+c)^(7/2)/a^2/c-17/48/a^2*x*(-a^2*c*x^2+c)^(5/2)-85/192*c*x*(-a^2*c*x^2+c)^(3/2)/a^2-85/128*c
^2*x*(-a^2*c*x^2+c)^(1/2)/a^2-85/128/a^2*c^3/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/7/a^
3*(-a^2*c*x^2+c)^(7/2)/c-2/5/a^3*(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(5/2)+1/2/a^2*c*(-(x-1/a)^2*a^2*c-2*a*c*(x-1
/a))^(3/2)*x+3/4/a^2*c^2*(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2)*x+3/4/a^2*c^3/(a^2*c)^(1/2)*arctan((a^2*c)^(1/
2)*x/(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2))

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maxima [A]  time = 0.67, size = 215, normalized size = 1.33 \[ -\frac {1}{13440} \, a {\left (\frac {4760 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{3}} - \frac {1680 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x}{a^{3} c} - \frac {770 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x}{a^{3}} - \frac {10080 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x}{a^{3}} + \frac {8925 \, \sqrt {-a^{2} c x^{2} + c} c^{2} x}{a^{3}} + \frac {8925 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{a^{4}} + \frac {5376 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{4}} - \frac {3840 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{a^{4} c} + \frac {20160 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{a^{4}} - \frac {10080 \, c^{4} \arcsin \left (a x - 2\right )}{a^{7} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2*(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

-1/13440*a*(4760*(-a^2*c*x^2 + c)^(5/2)*x/a^3 - 1680*(-a^2*c*x^2 + c)^(7/2)*x/(a^3*c) - 770*(-a^2*c*x^2 + c)^(
3/2)*c*x/a^3 - 10080*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^2*x/a^3 + 8925*sqrt(-a^2*c*x^2 + c)*c^2*x/a^3 + 8925*c^
(5/2)*arcsin(a*x)/a^4 + 5376*(-a^2*c*x^2 + c)^(5/2)/a^4 - 3840*(-a^2*c*x^2 + c)^(7/2)/(a^4*c) + 20160*sqrt(a^2
*c*x^2 - 4*a*c*x + 3*c)*c^2/a^4 - 10080*c^4*arcsin(a*x - 2)/(a^7*(-c/a^2)^(3/2)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {x^2\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(c - a^2*c*x^2)^(5/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

int(-(x^2*(c - a^2*c*x^2)^(5/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)

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sympy [C]  time = 21.28, size = 687, normalized size = 4.24 \[ - a^{4} c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{9}}{8 \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} x^{7}}{48 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{5}}{192 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{3}}{384 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i \sqrt {c} x}{128 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{128 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{9}}{8 \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} x^{7}}{48 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{5}}{192 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{3}}{384 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 \sqrt {c} x}{128 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} \operatorname {asin}{\left (a x \right )}}{128 a^{7}} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**(5/2),x)

[Out]

-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1))
- I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c
)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x*
*9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**
2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqr
t(c)*asin(a*x)/(128*a**7), True)) - 2*a**3*c**2*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x
**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0))
, (sqrt(c)*x**6/6, True)) + 2*a*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(1
5*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + c**2*Piecewise((I*a**2*sqrt
(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x*
*2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) +
3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3)
, True))

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