3.339 \(\int x^4 (A+B x) (a+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=198 \[ \frac {3 a^5 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}+\frac {3 a^4 A x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {a \left (a+c x^2\right )^{7/2} (640 a B-2079 A c x)}{55440 c^3}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c} \]

[Out]

1/128*a^3*A*x*(c*x^2+a)^(3/2)/c^2+1/160*a^2*A*x*(c*x^2+a)^(5/2)/c^2-4/99*a*B*x^2*(c*x^2+a)^(7/2)/c^2+1/10*A*x^
3*(c*x^2+a)^(7/2)/c+1/11*B*x^4*(c*x^2+a)^(7/2)/c+1/55440*a*(-2079*A*c*x+640*B*a)*(c*x^2+a)^(7/2)/c^3+3/256*a^5
*A*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(5/2)+3/256*a^4*A*x*(c*x^2+a)^(1/2)/c^2

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Rubi [A]  time = 0.15, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {833, 780, 195, 217, 206} \[ \frac {3 a^4 A x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {3 a^5 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}+\frac {a \left (a+c x^2\right )^{7/2} (640 a B-2079 A c x)}{55440 c^3}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*(a + c*x^2)^(5/2),x]

[Out]

(3*a^4*A*x*Sqrt[a + c*x^2])/(256*c^2) + (a^3*A*x*(a + c*x^2)^(3/2))/(128*c^2) + (a^2*A*x*(a + c*x^2)^(5/2))/(1
60*c^2) - (4*a*B*x^2*(a + c*x^2)^(7/2))/(99*c^2) + (A*x^3*(a + c*x^2)^(7/2))/(10*c) + (B*x^4*(a + c*x^2)^(7/2)
)/(11*c) + (a*(640*a*B - 2079*A*c*x)*(a + c*x^2)^(7/2))/(55440*c^3) + (3*a^5*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*
x^2]])/(256*c^(5/2))

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 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 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])

Rubi steps

\begin {align*} \int x^4 (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {\int x^3 (-4 a B+11 A c x) \left (a+c x^2\right )^{5/2} \, dx}{11 c}\\ &=\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {\int x^2 (-33 a A c-40 a B c x) \left (a+c x^2\right )^{5/2} \, dx}{110 c^2}\\ &=-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {\int x \left (80 a^2 B c-297 a A c^2 x\right ) \left (a+c x^2\right )^{5/2} \, dx}{990 c^3}\\ &=-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac {\left (3 a^2 A\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{80 c^2}\\ &=\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac {\left (a^3 A\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{32 c^2}\\ &=\frac {a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac {\left (3 a^4 A\right ) \int \sqrt {a+c x^2} \, dx}{128 c^2}\\ &=\frac {3 a^4 A x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac {\left (3 a^5 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{256 c^2}\\ &=\frac {3 a^4 A x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac {\left (3 a^5 A\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{256 c^2}\\ &=\frac {3 a^4 A x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac {A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac {a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac {3 a^5 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 151, normalized size = 0.76 \[ \frac {\sqrt {a+c x^2} \left (\frac {10395 a^{9/2} A \sqrt {c} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}+10240 a^5 B-5 a^4 c x (2079 A+1024 B x)+30 a^3 c^2 x^3 (231 A+128 B x)+8 a^2 c^3 x^5 (21483 A+18080 B x)+112 a c^4 x^7 (2079 A+1840 B x)+8064 c^5 x^9 (11 A+10 B x)\right )}{887040 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + c*x^2]*(10240*a^5*B + 8064*c^5*x^9*(11*A + 10*B*x) + 30*a^3*c^2*x^3*(231*A + 128*B*x) - 5*a^4*c*x*(2
079*A + 1024*B*x) + 112*a*c^4*x^7*(2079*A + 1840*B*x) + 8*a^2*c^3*x^5*(21483*A + 18080*B*x) + (10395*a^(9/2)*A
*Sqrt[c]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(887040*c^3)

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fricas [A]  time = 1.00, size = 320, normalized size = 1.62 \[ \left [\frac {10395 \, A a^{5} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (80640 \, B c^{5} x^{10} + 88704 \, A c^{5} x^{9} + 206080 \, B a c^{4} x^{8} + 232848 \, A a c^{4} x^{7} + 144640 \, B a^{2} c^{3} x^{6} + 171864 \, A a^{2} c^{3} x^{5} + 3840 \, B a^{3} c^{2} x^{4} + 6930 \, A a^{3} c^{2} x^{3} - 5120 \, B a^{4} c x^{2} - 10395 \, A a^{4} c x + 10240 \, B a^{5}\right )} \sqrt {c x^{2} + a}}{1774080 \, c^{3}}, -\frac {10395 \, A a^{5} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (80640 \, B c^{5} x^{10} + 88704 \, A c^{5} x^{9} + 206080 \, B a c^{4} x^{8} + 232848 \, A a c^{4} x^{7} + 144640 \, B a^{2} c^{3} x^{6} + 171864 \, A a^{2} c^{3} x^{5} + 3840 \, B a^{3} c^{2} x^{4} + 6930 \, A a^{3} c^{2} x^{3} - 5120 \, B a^{4} c x^{2} - 10395 \, A a^{4} c x + 10240 \, B a^{5}\right )} \sqrt {c x^{2} + a}}{887040 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

[1/1774080*(10395*A*a^5*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(80640*B*c^5*x^10 + 88704*
A*c^5*x^9 + 206080*B*a*c^4*x^8 + 232848*A*a*c^4*x^7 + 144640*B*a^2*c^3*x^6 + 171864*A*a^2*c^3*x^5 + 3840*B*a^3
*c^2*x^4 + 6930*A*a^3*c^2*x^3 - 5120*B*a^4*c*x^2 - 10395*A*a^4*c*x + 10240*B*a^5)*sqrt(c*x^2 + a))/c^3, -1/887
040*(10395*A*a^5*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (80640*B*c^5*x^10 + 88704*A*c^5*x^9 + 206080*B*
a*c^4*x^8 + 232848*A*a*c^4*x^7 + 144640*B*a^2*c^3*x^6 + 171864*A*a^2*c^3*x^5 + 3840*B*a^3*c^2*x^4 + 6930*A*a^3
*c^2*x^3 - 5120*B*a^4*c*x^2 - 10395*A*a^4*c*x + 10240*B*a^5)*sqrt(c*x^2 + a))/c^3]

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giac [A]  time = 0.21, size = 155, normalized size = 0.78 \[ -\frac {3 \, A a^{5} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{256 \, c^{\frac {5}{2}}} + \frac {1}{887040} \, {\left (\frac {10240 \, B a^{5}}{c^{3}} - {\left (\frac {10395 \, A a^{4}}{c^{2}} + 2 \, {\left (\frac {2560 \, B a^{4}}{c^{2}} - {\left (\frac {3465 \, A a^{3}}{c} + 4 \, {\left (\frac {480 \, B a^{3}}{c} + {\left (21483 \, A a^{2} + 2 \, {\left (9040 \, B a^{2} + 7 \, {\left (2079 \, A a c + 8 \, {\left (230 \, B a c + 9 \, {\left (10 \, B c^{2} x + 11 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

-3/256*A*a^5*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/887040*(10240*B*a^5/c^3 - (10395*A*a^4/c^2 + 2
*(2560*B*a^4/c^2 - (3465*A*a^3/c + 4*(480*B*a^3/c + (21483*A*a^2 + 2*(9040*B*a^2 + 7*(2079*A*a*c + 8*(230*B*a*
c + 9*(10*B*c^2*x + 11*A*c^2)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(c*x^2 + a)

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maple [A]  time = 0.06, size = 174, normalized size = 0.88 \[ \frac {3 A \,a^{5} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{256 c^{\frac {5}{2}}}+\frac {3 \sqrt {c \,x^{2}+a}\, A \,a^{4} x}{256 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,x^{4}}{11 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,a^{3} x}{128 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,x^{3}}{10 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,a^{2} x}{160 c^{2}}-\frac {4 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B a \,x^{2}}{99 c^{2}}-\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} A a x}{80 c^{2}}+\frac {8 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,a^{2}}{693 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(c*x^2+a)^(5/2),x)

[Out]

1/11*B*x^4*(c*x^2+a)^(7/2)/c-4/99*a*B*x^2*(c*x^2+a)^(7/2)/c^2+8/693*B*a^2/c^3*(c*x^2+a)^(7/2)+1/10*A*x^3*(c*x^
2+a)^(7/2)/c-3/80*A*a/c^2*x*(c*x^2+a)^(7/2)+1/160*a^2*A*x*(c*x^2+a)^(5/2)/c^2+1/128*a^3*A*x*(c*x^2+a)^(3/2)/c^
2+3/256*a^4*A*x*(c*x^2+a)^(1/2)/c^2+3/256*A*a^5/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))

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maxima [A]  time = 0.53, size = 166, normalized size = 0.84 \[ \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B x^{4}}{11 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A x^{3}}{10 \, c} - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B a x^{2}}{99 \, c^{2}} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A a x}{80 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A a^{2} x}{160 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A a^{3} x}{128 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} A a^{4} x}{256 \, c^{2}} + \frac {3 \, A a^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{256 \, c^{\frac {5}{2}}} + \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B a^{2}}{693 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

1/11*(c*x^2 + a)^(7/2)*B*x^4/c + 1/10*(c*x^2 + a)^(7/2)*A*x^3/c - 4/99*(c*x^2 + a)^(7/2)*B*a*x^2/c^2 - 3/80*(c
*x^2 + a)^(7/2)*A*a*x/c^2 + 1/160*(c*x^2 + a)^(5/2)*A*a^2*x/c^2 + 1/128*(c*x^2 + a)^(3/2)*A*a^3*x/c^2 + 3/256*
sqrt(c*x^2 + a)*A*a^4*x/c^2 + 3/256*A*a^5*arcsinh(c*x/sqrt(a*c))/c^(5/2) + 8/693*(c*x^2 + a)^(7/2)*B*a^2/c^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + c*x^2)^(5/2)*(A + B*x),x)

[Out]

int(x^4*(a + c*x^2)^(5/2)*(A + B*x), x)

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sympy [A]  time = 37.23, size = 541, normalized size = 2.73 \[ - \frac {3 A a^{\frac {9}{2}} x}{256 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a^{\frac {7}{2}} x^{3}}{256 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {129 A a^{\frac {5}{2}} x^{5}}{640 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {73 A a^{\frac {3}{2}} c x^{7}}{160 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {29 A \sqrt {a} c^{2} x^{9}}{80 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A a^{5} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{256 c^{\frac {5}{2}}} + \frac {A c^{3} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B a^{2} \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 B a c \left (\begin {cases} - \frac {16 a^{4} \sqrt {a + c x^{2}}}{315 c^{4}} + \frac {8 a^{3} x^{2} \sqrt {a + c x^{2}}}{315 c^{3}} - \frac {2 a^{2} x^{4} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{6} \sqrt {a + c x^{2}}}{63 c} + \frac {x^{8} \sqrt {a + c x^{2}}}{9} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + B c^{2} \left (\begin {cases} \frac {128 a^{5} \sqrt {a + c x^{2}}}{3465 c^{5}} - \frac {64 a^{4} x^{2} \sqrt {a + c x^{2}}}{3465 c^{4}} + \frac {16 a^{3} x^{4} \sqrt {a + c x^{2}}}{1155 c^{3}} - \frac {8 a^{2} x^{6} \sqrt {a + c x^{2}}}{693 c^{2}} + \frac {a x^{8} \sqrt {a + c x^{2}}}{99 c} + \frac {x^{10} \sqrt {a + c x^{2}}}{11} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{10}}{10} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(c*x**2+a)**(5/2),x)

[Out]

-3*A*a**(9/2)*x/(256*c**2*sqrt(1 + c*x**2/a)) - A*a**(7/2)*x**3/(256*c*sqrt(1 + c*x**2/a)) + 129*A*a**(5/2)*x*
*5/(640*sqrt(1 + c*x**2/a)) + 73*A*a**(3/2)*c*x**7/(160*sqrt(1 + c*x**2/a)) + 29*A*sqrt(a)*c**2*x**9/(80*sqrt(
1 + c*x**2/a)) + 3*A*a**5*asinh(sqrt(c)*x/sqrt(a))/(256*c**(5/2)) + A*c**3*x**11/(10*sqrt(a)*sqrt(1 + c*x**2/a
)) + B*a**2*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*s
qrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + 2*B*a*c*Piecewise((-16*
a**4*sqrt(a + c*x**2)/(315*c**4) + 8*a**3*x**2*sqrt(a + c*x**2)/(315*c**3) - 2*a**2*x**4*sqrt(a + c*x**2)/(105
*c**2) + a*x**6*sqrt(a + c*x**2)/(63*c) + x**8*sqrt(a + c*x**2)/9, Ne(c, 0)), (sqrt(a)*x**8/8, True)) + B*c**2
*Piecewise((128*a**5*sqrt(a + c*x**2)/(3465*c**5) - 64*a**4*x**2*sqrt(a + c*x**2)/(3465*c**4) + 16*a**3*x**4*s
qrt(a + c*x**2)/(1155*c**3) - 8*a**2*x**6*sqrt(a + c*x**2)/(693*c**2) + a*x**8*sqrt(a + c*x**2)/(99*c) + x**10
*sqrt(a + c*x**2)/11, Ne(c, 0)), (sqrt(a)*x**10/10, True))

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