3.3.37 \(\int x^2 (c (a+b x^2)^3)^{3/2} \, dx\) [237]

Optimal. Leaf size=253 \[ \frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {21 a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1024 b^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/2}} \]

[Out]

7/128*a^3*c*x^3*(c*(b*x^2+a)^3)^(1/2)+21/1024*a^5*c*x*(c*(b*x^2+a)^3)^(1/2)/b/(b*x^2+a)+21/512*a^4*c*x^3*(c*(b
*x^2+a)^3)^(1/2)/(b*x^2+a)+21/320*a^2*c*x^3*(b*x^2+a)*(c*(b*x^2+a)^3)^(1/2)+3/40*a*c*x^3*(b*x^2+a)^2*(c*(b*x^2
+a)^3)^(1/2)+1/12*c*x^3*(b*x^2+a)^3*(c*(b*x^2+a)^3)^(1/2)-21/1024*a^(9/2)*c*arcsinh(x*b^(1/2)/a^(1/2))*(c*(b*x
^2+a)^3)^(1/2)/b^(3/2)/(1+b*x^2/a)^(3/2)

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Rubi [A]
time = 0.09, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1973, 285, 327, 221} \begin {gather*} -\frac {21 a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1024 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(c*(a + b*x^2)^3)^(3/2),x]

[Out]

(7*a^3*c*x^3*Sqrt[c*(a + b*x^2)^3])/128 + (21*a^5*c*x*Sqrt[c*(a + b*x^2)^3])/(1024*b*(a + b*x^2)) + (21*a^4*c*
x^3*Sqrt[c*(a + b*x^2)^3])/(512*(a + b*x^2)) + (21*a^2*c*x^3*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/320 + (3*a*c*x
^3*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/40 + (c*x^3*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/12 - (21*a^(9/2)*c*Sq
rt[c*(a + b*x^2)^3]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(1024*b^(3/2)*(1 + (b*x^2)/a)^(3/2))

Rule 221

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

Rule 285

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 327

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 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

Rubi steps

\begin {align*} \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{9/2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (3 a c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{7/2} \, dx}{4 \left (a+b x^2\right )^{3/2}}\\ &=\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^2 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{5/2} \, dx}{40 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{3/2} \, dx}{64 \left (a+b x^2\right )^{3/2}}\\ &=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^4 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \sqrt {a+b x^2} \, dx}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^5 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{512 \left (a+b x^2\right )^{3/2}}\\ &=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {\left (21 a^6 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{1024 b \left (a+b x^2\right )^{3/2}}\\ &=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {\left (21 a^6 c \sqrt {c \left (a+b x^2\right )^3}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{1024 b \left (a+b x^2\right )^{3/2}}\\ &=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {21 a^6 c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 134, normalized size = 0.53 \begin {gather*} \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {b} x \sqrt {a+b x^2} \left (315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right )+315 a^6 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{15360 b^{3/2} \left (a+b x^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(c*(a + b*x^2)^3)^(3/2),x]

[Out]

((c*(a + b*x^2)^3)^(3/2)*(Sqrt[b]*x*Sqrt[a + b*x^2]*(315*a^5 + 4910*a^4*b*x^2 + 11432*a^3*b^2*x^4 + 12144*a^2*
b^3*x^6 + 6272*a*b^4*x^8 + 1280*b^5*x^10) + 315*a^6*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]))/(15360*b^(3/2)*(a +
b*x^2)^(9/2))

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Maple [A]
time = 0.06, size = 236, normalized size = 0.93

method	result	size
risch	\(\frac {x \left (1280 b^{5} x^{10}+6272 b^{4} a \,x^{8}+12144 a^{2} b^{3} x^{6}+11432 b^{2} a^{3} x^{4}+4910 b \,a^{4} x^{2}+315 a^{5}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}}{15360 \left (b \,x^{2}+a \right ) b}-\frac {21 a^{6} \ln \left (\frac {b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}\, \sqrt {c \left (b \,x^{2}+a \right )}}{1024 b \sqrt {b c}\, \left (b \,x^{2}+a \right )^{2}}\)	\(155\)
default	\(\frac {\left (c \left (b \,x^{2}+a \right )^{3}\right )^{\frac {3}{2}} \left (1280 x^{7} \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} b^{3} \sqrt {b c}+3712 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a \,b^{2} x^{5}+3440 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{2} b \,x^{3}+840 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{3} x -210 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{4} c x -315 \sqrt {b c}\, \sqrt {b c \,x^{2}+a c}\, a^{5} c^{2} x -315 \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right ) a^{6} c^{3}\right )}{15360 b \left (b \,x^{2}+a \right )^{3} \left (c \left (b \,x^{2}+a \right )\right )^{\frac {3}{2}} c \sqrt {b c}}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15360*(c*(b*x^2+a)^3)^(3/2)/b*(1280*x^7*(b*c*x^2+a*c)^(5/2)*b^3*(b*c)^(1/2)+3712*(b*c)^(1/2)*(b*c*x^2+a*c)^(
5/2)*a*b^2*x^5+3440*(b*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*a^2*b*x^3+840*(b*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*a^3*x-210*(b
*c)^(1/2)*(b*c*x^2+a*c)^(3/2)*a^4*c*x-315*(b*c)^(1/2)*(b*c*x^2+a*c)^(1/2)*a^5*c^2*x-315*ln((b*c*x+(b*c*x^2+a*c
)^(1/2)*(b*c)^(1/2))/(b*c)^(1/2))*a^6*c^3)/(b*x^2+a)^3/(c*(b*x^2+a))^(3/2)/c/(b*c)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((b*x^2 + a)^3*c)^(3/2)*x^2, x)

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Fricas [A]
time = 0.41, size = 433, normalized size = 1.71 \begin {gather*} \left [\frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {\frac {c}{b}} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c - 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {\frac {c}{b}}}{b x^{2} + a}\right ) + 2 \, {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{30720 \, {\left (b^{2} x^{2} + a b\right )}}, \frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {-\frac {c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) + {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{15360 \, {\left (b^{2} x^{2} + a b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30720*(315*(a^6*b*c*x^2 + a^7*c)*sqrt(c/b)*log(-(2*b^2*c*x^4 + 3*a*b*c*x^2 + a^2*c - 2*sqrt(b^3*c*x^6 + 3*a
*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*b*x*sqrt(c/b))/(b*x^2 + a)) + 2*(1280*b^5*c*x^11 + 6272*a*b^4*c*x^9 + 1214
4*a^2*b^3*c*x^7 + 11432*a^3*b^2*c*x^5 + 4910*a^4*b*c*x^3 + 315*a^5*c*x)*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2
*b*c*x^2 + a^3*c))/(b^2*x^2 + a*b), 1/15360*(315*(a^6*b*c*x^2 + a^7*c)*sqrt(-c/b)*arctan(sqrt(b^3*c*x^6 + 3*a*
b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*b*x*sqrt(-c/b)/(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)) + (1280*b^5*c*x^11 + 6272
*a*b^4*c*x^9 + 12144*a^2*b^3*c*x^7 + 11432*a^3*b^2*c*x^5 + 4910*a^4*b*c*x^3 + 315*a^5*c*x)*sqrt(b^3*c*x^6 + 3*
a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b^2*x^2 + a*b)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (c \left (a + b x^{2}\right )^{3}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(c*(a + b*x**2)**3)**(3/2), x)

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Giac [A]
time = 4.76, size = 177, normalized size = 0.70 \begin {gather*} \frac {1}{15360} \, {\left (\frac {315 \, a^{6} c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b} + {\left (\frac {315 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{b} + 2 \, {\left (2455 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (1429 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (759 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 8 \, {\left (10 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 49 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15360*(315*a^6*c*log(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sgn(b*x^2 + a)/(sqrt(b*c)*b) + (315*a^5*sgn(b*
x^2 + a)/b + 2*(2455*a^4*sgn(b*x^2 + a) + 4*(1429*a^3*b*sgn(b*x^2 + a) + 2*(759*a^2*b^2*sgn(b*x^2 + a) + 8*(10
*b^4*x^2*sgn(b*x^2 + a) + 49*a*b^3*sgn(b*x^2 + a))*x^2)*x^2)*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*x)*c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*(a + b*x^2)^3)^(3/2),x)

[Out]

int(x^2*(c*(a + b*x^2)^3)^(3/2), x)

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