3.244 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx\)

Optimal. Leaf size=106 \[ \frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{315 a (c-a c x)^{5/2}}+\frac {16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt {c-a c x}} \]

[Out]

64/315*c^6*(-a^2*x^2+1)^(5/2)/a/(-a*c*x+c)^(5/2)+16/63*c^5*(-a^2*x^2+1)^(5/2)/a/(-a*c*x+c)^(3/2)+2/9*c^4*(-a^2
*x^2+1)^(5/2)/a/(-a*c*x+c)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6127, 657, 649} \[ \frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{315 a (c-a c x)^{5/2}}+\frac {16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a*c*x)^(7/2),x]

[Out]

(64*c^6*(1 - a^2*x^2)^(5/2))/(315*a*(c - a*c*x)^(5/2)) + (16*c^5*(1 - a^2*x^2)^(5/2))/(63*a*(c - a*c*x)^(3/2))
+ (2*c^4*(1 - a^2*x^2)^(5/2))/(9*a*Sqrt[c - a*c*x])

Rule 649

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 657

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*Simplify[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, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

Rule 6127

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

Rubi steps

\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=c^3 \int \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt {c-a c x}}+\frac {1}{9} \left (8 c^4\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt {c-a c x}}+\frac {1}{63} \left (32 c^5\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {64 c^6 \left (1-a^2 x^2\right )^{5/2}}{315 a (c-a c x)^{5/2}}+\frac {16 c^5 \left (1-a^2 x^2\right )^{5/2}}{63 a (c-a c x)^{3/2}}+\frac {2 c^4 \left (1-a^2 x^2\right )^{5/2}}{9 a \sqrt {c-a c x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 54, normalized size = 0.51 \[ \frac {2 c^3 (a x+1)^{5/2} \left (35 a^2 x^2-110 a x+107\right ) \sqrt {c-a c x}}{315 a \sqrt {1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a*c*x)^(7/2),x]

[Out]

(2*c^3*(1 + a*x)^(5/2)*Sqrt[c - a*c*x]*(107 - 110*a*x + 35*a^2*x^2))/(315*a*Sqrt[1 - a*x])

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fricas [A]  time = 0.48, size = 80, normalized size = 0.75 \[ -\frac {2 \, {\left (35 \, a^{4} c^{3} x^{4} - 40 \, a^{3} c^{3} x^{3} - 78 \, a^{2} c^{3} x^{2} + 104 \, a c^{3} x + 107 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{315 \, {\left (a^{2} x - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="fricas")

[Out]

-2/315*(35*a^4*c^3*x^4 - 40*a^3*c^3*x^3 - 78*a^2*c^3*x^2 + 104*a*c^3*x + 107*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c
*x + c)/(a^2*x - a)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.03, size = 55, normalized size = 0.52 \[ \frac {2 \left (a x +1\right )^{4} \left (35 a^{2} x^{2}-110 a x +107\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x)

[Out]

2/315*(a*x+1)^4*(35*a^2*x^2-110*a*x+107)*(-a*c*x+c)^(7/2)/a/(a*x-1)^2/(-a^2*x^2+1)^(3/2)

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maxima [B]  time = 0.38, size = 210, normalized size = 1.98 \[ \frac {2 \, {\left (5 \, a^{5} c^{\frac {7}{2}} x^{5} - 20 \, a^{4} c^{\frac {7}{2}} x^{4} + 41 \, a^{3} c^{\frac {7}{2}} x^{3} - 82 \, a^{2} c^{\frac {7}{2}} x^{2} + 328 \, a c^{\frac {7}{2}} x + 656 \, c^{\frac {7}{2}}\right )}}{45 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (15 \, a^{4} c^{\frac {7}{2}} x^{4} - 66 \, a^{3} c^{\frac {7}{2}} x^{3} + 167 \, a^{2} c^{\frac {7}{2}} x^{2} - 668 \, a c^{\frac {7}{2}} x - 1336 \, c^{\frac {7}{2}}\right )}}{35 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (3 \, a^{3} c^{\frac {7}{2}} x^{3} - 16 \, a^{2} c^{\frac {7}{2}} x^{2} + 79 \, a c^{\frac {7}{2}} x + 158 \, c^{\frac {7}{2}}\right )}}{5 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (a^{2} c^{\frac {7}{2}} x^{2} - 10 \, a c^{\frac {7}{2}} x - 23 \, c^{\frac {7}{2}}\right )}}{3 \, \sqrt {a x + 1} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="maxima")

[Out]

2/45*(5*a^5*c^(7/2)*x^5 - 20*a^4*c^(7/2)*x^4 + 41*a^3*c^(7/2)*x^3 - 82*a^2*c^(7/2)*x^2 + 328*a*c^(7/2)*x + 656
*c^(7/2))/(sqrt(a*x + 1)*a) + 2/35*(15*a^4*c^(7/2)*x^4 - 66*a^3*c^(7/2)*x^3 + 167*a^2*c^(7/2)*x^2 - 668*a*c^(7
/2)*x - 1336*c^(7/2))/(sqrt(a*x + 1)*a) + 2/5*(3*a^3*c^(7/2)*x^3 - 16*a^2*c^(7/2)*x^2 + 79*a*c^(7/2)*x + 158*c
^(7/2))/(sqrt(a*x + 1)*a) + 2/3*(a^2*c^(7/2)*x^2 - 10*a*c^(7/2)*x - 23*c^(7/2))/(sqrt(a*x + 1)*a)

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mupad [B]  time = 0.98, size = 79, normalized size = 0.75 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {422\,c^3\,x}{315}+\frac {214\,c^3}{315\,a}+\frac {52\,a\,c^3\,x^2}{315}-\frac {236\,a^2\,c^3\,x^3}{315}-\frac {2\,a^3\,c^3\,x^4}{63}+\frac {2\,a^4\,c^3\,x^5}{9}\right )}{\sqrt {1-a^2\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^(7/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)

[Out]

((c - a*c*x)^(1/2)*((422*c^3*x)/315 + (214*c^3)/(315*a) + (52*a*c^3*x^2)/315 - (236*a^2*c^3*x^3)/315 - (2*a^3*
c^3*x^4)/63 + (2*a^4*c^3*x^5)/9))/(1 - a^2*x^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(7/2),x)

[Out]

Integral((-c*(a*x - 1))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)

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