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

Optimal. Leaf size=221 \[ \frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}}-\frac {a^4 x \sqrt {a+b x^2} (12 A b-5 a B)}{1024 b^3}+\frac {a^3 x^3 \sqrt {a+b x^2} (12 A b-5 a B)}{1536 b^2}+\frac {a^2 x^5 \sqrt {a+b x^2} (12 A b-5 a B)}{384 b}+\frac {a x^5 \left (a+b x^2\right )^{3/2} (12 A b-5 a B)}{192 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 A b-5 a B)}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b} \]

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Rubi [A]  time = 0.10, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {459, 279, 321, 217, 206} \begin {gather*} -\frac {a^4 x \sqrt {a+b x^2} (12 A b-5 a B)}{1024 b^3}+\frac {a^3 x^3 \sqrt {a+b x^2} (12 A b-5 a B)}{1536 b^2}+\frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}}+\frac {a^2 x^5 \sqrt {a+b x^2} (12 A b-5 a B)}{384 b}+\frac {a x^5 \left (a+b x^2\right )^{3/2} (12 A b-5 a B)}{192 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 A b-5 a B)}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4*(12*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(1024*b^3) + (a^3*(12*A*b - 5*a*B)*x^3*Sqrt[a + b*x^2])/(1536*b^2) +
(a^2*(12*A*b - 5*a*B)*x^5*Sqrt[a + b*x^2])/(384*b) + (a*(12*A*b - 5*a*B)*x^5*(a + b*x^2)^(3/2))/(192*b) + ((1
2*A*b - 5*a*B)*x^5*(a + b*x^2)^(5/2))/(120*b) + (B*x^5*(a + b*x^2)^(7/2))/(12*b) + (a^5*(12*A*b - 5*a*B)*ArcTa
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 459

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

Rubi steps

\begin {align*} \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx &=\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}-\frac {(-12 A b+5 a B) \int x^4 \left (a+b x^2\right )^{5/2} \, dx}{12 b}\\ &=\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {(a (12 A b-5 a B)) \int x^4 \left (a+b x^2\right )^{3/2} \, dx}{24 b}\\ &=\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^2 (12 A b-5 a B)\right ) \int x^4 \sqrt {a+b x^2} \, dx}{64 b}\\ &=\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^3 (12 A b-5 a B)\right ) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{384 b}\\ &=\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}-\frac {\left (a^4 (12 A b-5 a B)\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{512 b^2}\\ &=-\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{1024 b^3}\\ &=-\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{1024 b^3}\\ &=-\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 172, normalized size = 0.78 \begin {gather*} \frac {\sqrt {a+b x^2} \left (\sqrt {b} x \left (75 a^5 B-10 a^4 b \left (18 A+5 B x^2\right )+40 a^3 b^2 x^2 \left (3 A+B x^2\right )+48 a^2 b^3 x^4 \left (62 A+45 B x^2\right )+64 a b^4 x^6 \left (63 A+50 B x^2\right )+256 b^5 x^8 \left (6 A+5 B x^2\right )\right )-\frac {15 a^{9/2} (5 a B-12 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{15360 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(75*a^5*B + 40*a^3*b^2*x^2*(3*A + B*x^2) + 256*b^5*x^8*(6*A + 5*B*x^2) - 10*a^4*b*
(18*A + 5*B*x^2) + 48*a^2*b^3*x^4*(62*A + 45*B*x^2) + 64*a*b^4*x^6*(63*A + 50*B*x^2)) - (15*a^(9/2)*(-12*A*b +
5*a*B)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(15360*b^(7/2))

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IntegrateAlgebraic [A]  time = 0.29, size = 175, normalized size = 0.79 \begin {gather*} \frac {\left (5 a^6 B-12 a^5 A b\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{1024 b^{7/2}}+\frac {\sqrt {a+b x^2} \left (75 a^5 B x-180 a^4 A b x-50 a^4 b B x^3+120 a^3 A b^2 x^3+40 a^3 b^2 B x^5+2976 a^2 A b^3 x^5+2160 a^2 b^3 B x^7+4032 a A b^4 x^7+3200 a b^4 B x^9+1536 A b^5 x^9+1280 b^5 B x^{11}\right )}{15360 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^2]*(-180*a^4*A*b*x + 75*a^5*B*x + 120*a^3*A*b^2*x^3 - 50*a^4*b*B*x^3 + 2976*a^2*A*b^3*x^5 + 40*a
^3*b^2*B*x^5 + 4032*a*A*b^4*x^7 + 2160*a^2*b^3*B*x^7 + 1536*A*b^5*x^9 + 3200*a*b^4*B*x^9 + 1280*b^5*B*x^11))/(
15360*b^3) + ((-12*a^5*A*b + 5*a^6*B)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(1024*b^(7/2))

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fricas [A]  time = 1.06, size = 355, normalized size = 1.61 \begin {gather*} \left [-\frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (1280 \, B b^{6} x^{11} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{9} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{7} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{5} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} + 15 \, {\left (5 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{30720 \, b^{4}}, \frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (1280 \, B b^{6} x^{11} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{9} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{7} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{5} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} + 15 \, {\left (5 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{15360 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30720*(15*(5*B*a^6 - 12*A*a^5*b)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(1280*B*b^6*x
^11 + 128*(25*B*a*b^5 + 12*A*b^6)*x^9 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^7 + 8*(5*B*a^3*b^3 + 372*A*a^2*b^4)*
x^5 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x^3 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*x)*sqrt(b*x^2 + a))/b^4, 1/15360*(15
*(5*B*a^6 - 12*A*a^5*b)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (1280*B*b^6*x^11 + 128*(25*B*a*b^5 + 12*
A*b^6)*x^9 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^7 + 8*(5*B*a^3*b^3 + 372*A*a^2*b^4)*x^5 - 10*(5*B*a^4*b^2 - 12*
A*a^3*b^3)*x^3 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*x)*sqrt(b*x^2 + a))/b^4]

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giac [A]  time = 0.35, size = 195, normalized size = 0.88 \begin {gather*} \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B b^{2} x^{2} + \frac {25 \, B a b^{11} + 12 \, A b^{12}}{b^{10}}\right )} x^{2} + \frac {9 \, {\left (15 \, B a^{2} b^{10} + 28 \, A a b^{11}\right )}}{b^{10}}\right )} x^{2} + \frac {5 \, B a^{3} b^{9} + 372 \, A a^{2} b^{10}}{b^{10}}\right )} x^{2} - \frac {5 \, {\left (5 \, B a^{4} b^{8} - 12 \, A a^{3} b^{9}\right )}}{b^{10}}\right )} x^{2} + \frac {15 \, {\left (5 \, B a^{5} b^{7} - 12 \, A a^{4} b^{8}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15360*(2*(4*(2*(8*(10*B*b^2*x^2 + (25*B*a*b^11 + 12*A*b^12)/b^10)*x^2 + 9*(15*B*a^2*b^10 + 28*A*a*b^11)/b^10
)*x^2 + (5*B*a^3*b^9 + 372*A*a^2*b^10)/b^10)*x^2 - 5*(5*B*a^4*b^8 - 12*A*a^3*b^9)/b^10)*x^2 + 15*(5*B*a^5*b^7
- 12*A*a^4*b^8)/b^10)*sqrt(b*x^2 + a)*x + 1/1024*(5*B*a^6 - 12*A*a^5*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))
/b^(7/2)

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maple [A]  time = 0.02, size = 257, normalized size = 1.16 \begin {gather*} \frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B \,x^{5}}{12 b}+\frac {3 A \,a^{5} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}-\frac {5 B \,a^{6} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {7}{2}}}+\frac {3 \sqrt {b \,x^{2}+a}\, A \,a^{4} x}{256 b^{2}}-\frac {5 \sqrt {b \,x^{2}+a}\, B \,a^{5} x}{1024 b^{3}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,a^{3} x}{128 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A \,x^{3}}{10 b}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{4} x}{1536 b^{3}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B a \,x^{3}}{24 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,a^{2} x}{160 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B \,a^{3} x}{384 b^{3}}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} A a x}{80 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B \,a^{2} x}{64 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*B*x^5*(b*x^2+a)^(7/2)/b-1/24*B*a/b^2*x^3*(b*x^2+a)^(7/2)+1/64*B*a^2/b^3*x*(b*x^2+a)^(7/2)-1/384*B*a^3/b^3
*x*(b*x^2+a)^(5/2)-5/1536*B*a^4/b^3*x*(b*x^2+a)^(3/2)-5/1024*B*a^5/b^3*x*(b*x^2+a)^(1/2)-5/1024*B*a^6/b^(7/2)*
ln(b^(1/2)*x+(b*x^2+a)^(1/2))+1/10*A*x^3*(b*x^2+a)^(7/2)/b-3/80*A*a/b^2*x*(b*x^2+a)^(7/2)+1/160*A*a^2/b^2*x*(b
*x^2+a)^(5/2)+1/128*A*a^3/b^2*x*(b*x^2+a)^(3/2)+3/256*A*a^4/b^2*x*(b*x^2+a)^(1/2)+3/256*A*a^5/b^(5/2)*ln(b^(1/
2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.12, size = 242, normalized size = 1.10 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{5}}{12 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x^{3}}{24 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x^{3}}{10 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} x}{64 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3} x}{384 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{4} x}{1536 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} B a^{5} x}{1024 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} A a^{4} x}{256 \, b^{2}} - \frac {5 \, B a^{6} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b*x^2 + a)^(7/2)*B*x^5/b - 1/24*(b*x^2 + a)^(7/2)*B*a*x^3/b^2 + 1/10*(b*x^2 + a)^(7/2)*A*x^3/b + 1/64*(b
*x^2 + a)^(7/2)*B*a^2*x/b^3 - 1/384*(b*x^2 + a)^(5/2)*B*a^3*x/b^3 - 5/1536*(b*x^2 + a)^(3/2)*B*a^4*x/b^3 - 5/1
024*sqrt(b*x^2 + a)*B*a^5*x/b^3 - 3/80*(b*x^2 + a)^(7/2)*A*a*x/b^2 + 1/160*(b*x^2 + a)^(5/2)*A*a^2*x/b^2 + 1/1
28*(b*x^2 + a)^(3/2)*A*a^3*x/b^2 + 3/256*sqrt(b*x^2 + a)*A*a^4*x/b^2 - 5/1024*B*a^6*arcsinh(b*x/sqrt(a*b))/b^(
7/2) + 3/256*A*a^5*arcsinh(b*x/sqrt(a*b))/b^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 83.28, size = 405, normalized size = 1.83 \begin {gather*} - \frac {3 A a^{\frac {9}{2}} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A a^{\frac {7}{2}} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {129 A a^{\frac {5}{2}} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {73 A a^{\frac {3}{2}} b x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 A \sqrt {a} b^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} + \frac {A b^{3} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 B a^{\frac {11}{2}} x}{1024 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 B a^{\frac {9}{2}} x^{3}}{3072 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {7}{2}} x^{5}}{1536 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {55 B a^{\frac {5}{2}} x^{7}}{384 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {67 B a^{\frac {3}{2}} b x^{9}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 B \sqrt {a} b^{2} x^{11}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 B a^{6} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{1024 b^{\frac {7}{2}}} + \frac {B b^{3} x^{13}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*A*a**(9/2)*x/(256*b**2*sqrt(1 + b*x**2/a)) - A*a**(7/2)*x**3/(256*b*sqrt(1 + b*x**2/a)) + 129*A*a**(5/2)*x*
*5/(640*sqrt(1 + b*x**2/a)) + 73*A*a**(3/2)*b*x**7/(160*sqrt(1 + b*x**2/a)) + 29*A*sqrt(a)*b**2*x**9/(80*sqrt(
1 + b*x**2/a)) + 3*A*a**5*asinh(sqrt(b)*x/sqrt(a))/(256*b**(5/2)) + A*b**3*x**11/(10*sqrt(a)*sqrt(1 + b*x**2/a
)) + 5*B*a**(11/2)*x/(1024*b**3*sqrt(1 + b*x**2/a)) + 5*B*a**(9/2)*x**3/(3072*b**2*sqrt(1 + b*x**2/a)) - B*a**
(7/2)*x**5/(1536*b*sqrt(1 + b*x**2/a)) + 55*B*a**(5/2)*x**7/(384*sqrt(1 + b*x**2/a)) + 67*B*a**(3/2)*b*x**9/(1
92*sqrt(1 + b*x**2/a)) + 7*B*sqrt(a)*b**2*x**11/(24*sqrt(1 + b*x**2/a)) - 5*B*a**6*asinh(sqrt(b)*x/sqrt(a))/(1
024*b**(7/2)) + B*b**3*x**13/(12*sqrt(a)*sqrt(1 + b*x**2/a))

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