3.3.41 \(\int \frac {(c (a+b x^2)^3)^{3/2}}{x^2} \, dx\) [241]

Optimal. Leaf size=208 \[ \frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {315 a^{5/2} \sqrt {b} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 \left (1+\frac {b x^2}{a}\right )^{3/2}} \]

[Out]

105/64*a^2*b*c*x*(c*(b*x^2+a)^3)^(1/2)+315/128*a^3*b*c*x*(c*(b*x^2+a)^3)^(1/2)/(b*x^2+a)+21/16*a*b*c*x*(b*x^2+
a)*(c*(b*x^2+a)^3)^(1/2)+9/8*b*c*x*(b*x^2+a)^2*(c*(b*x^2+a)^3)^(1/2)-c*(b*x^2+a)^3*(c*(b*x^2+a)^3)^(1/2)/x+315
/128*a^(5/2)*c*arcsinh(x*b^(1/2)/a^(1/2))*b^(1/2)*(c*(b*x^2+a)^3)^(1/2)/(1+b*x^2/a)^(3/2)

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Rubi [A]
time = 0.06, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1973, 283, 201, 221} \begin {gather*} \frac {315 a^{5/2} \sqrt {b} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 \left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x^2)^3)^(3/2)/x^2,x]

[Out]

(105*a^2*b*c*x*Sqrt[c*(a + b*x^2)^3])/64 + (315*a^3*b*c*x*Sqrt[c*(a + b*x^2)^3])/(128*(a + b*x^2)) + (21*a*b*c
*x*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/16 + (9*b*c*x*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/8 - (c*(a + b*x^2)^3*
Sqrt[c*(a + b*x^2)^3])/x + (315*a^(5/2)*Sqrt[b]*c*Sqrt[c*(a + b*x^2)^3]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(128*(1
+ (b*x^2)/a)^(3/2))

Rule 201

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 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 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 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 \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2}}{x^2} \, dx &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (9 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{7/2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (63 a b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{8 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (105 a^2 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{16 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (315 a^3 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \sqrt {a+b x^2} \, dx}{64 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (315 a^4 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (315 a^4 b 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 )}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {315 a^4 \sqrt {b} c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 \left (a+b x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 121, normalized size = 0.58 \begin {gather*} -\frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {a+b x^2} \left (128 a^4-325 a^3 b x^2-210 a^2 b^2 x^4-88 a b^3 x^6-16 b^4 x^8\right )+315 a^4 \sqrt {b} x \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{128 x \left (a+b x^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((c*(a + b*x^2)^3)^(3/2)*(Sqrt[a + b*x^2]*(128*a^4 - 325*a^3*b*x^2 - 210*a^2*b^2*x^4 - 88*a*b^3*x^6 - 1
6*b^4*x^8) + 315*a^4*Sqrt[b]*x*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]))/(x*(a + b*x^2)^(9/2))

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Maple [A]
time = 0.05, size = 215, normalized size = 1.03

method	result	size
risch	\(-\frac {\left (-16 b^{4} x^{8}-88 a \,b^{3} x^{6}-210 a^{2} b^{2} x^{4}-325 a^{3} b \,x^{2}+128 a^{4}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}}{128 \left (b \,x^{2}+a \right ) x}+\frac {315 b \,a^{4} \ln \left (\frac {b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}\, \sqrt {c \left (b \,x^{2}+a \right )}}{128 \sqrt {b c}\, \left (b \,x^{2}+a \right )^{2}}\)	\(141\)
default	\(-\frac {\left (c \left (b \,x^{2}+a \right )^{3}\right )^{\frac {3}{2}} \left (-16 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, b^{2} x^{4}-56 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a b \,x^{2}-210 \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b c}\, a^{2} b c \,x^{2}-315 \sqrt {b c \,x^{2}+a c}\, \sqrt {b c}\, a^{3} b \,c^{2} x^{2}-315 \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right ) a^{4} b \,c^{3} x +128 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a^{2}\right )}{128 \left (b \,x^{2}+a \right )^{3} \left (c \left (b \,x^{2}+a \right )\right )^{\frac {3}{2}} c \sqrt {b c}\, x}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(c*(b*x^2+a)^3)^(3/2)*(-16*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)*b^2*x^4-56*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)*a
*b*x^2-210*(b*c*x^2+a*c)^(3/2)*(b*c)^(1/2)*a^2*b*c*x^2-315*(b*c*x^2+a*c)^(1/2)*(b*c)^(1/2)*a^3*b*c^2*x^2-315*l
n((b*c*x+(b*c*x^2+a*c)^(1/2)*(b*c)^(1/2))/(b*c)^(1/2))*a^4*b*c^3*x+128*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)*a^2)/(b
*x^2+a)^3/(c*(b*x^2+a))^(3/2)/c/(b*c)^(1/2)/x

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

[Out]

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

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Fricas [A]
time = 0.38, size = 396, normalized size = 1.90 \begin {gather*} \left [\frac {315 \, {\left (a^{4} b c x^{3} + a^{5} c x\right )} \sqrt {b c} \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} \sqrt {b c} x}{b x^{2} + a}\right ) + 2 \, {\left (16 \, b^{4} c x^{8} + 88 \, a b^{3} c x^{6} + 210 \, a^{2} b^{2} c x^{4} + 325 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{256 \, {\left (b x^{3} + a x\right )}}, -\frac {315 \, {\left (a^{4} b c x^{3} + a^{5} c x\right )} \sqrt {-b c} \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} \sqrt {-b c} x}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) - {\left (16 \, b^{4} c x^{8} + 88 \, a b^{3} c x^{6} + 210 \, a^{2} b^{2} c x^{4} + 325 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{128 \, {\left (b x^{3} + a x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/256*(315*(a^4*b*c*x^3 + a^5*c*x)*sqrt(b*c)*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)*sqrt(b*c)*x)/(b*x^2 + a)) + 2*(16*b^4*c*x^8 + 88*a*b^3*c*x^6 + 210*a^2*b^2
*c*x^4 + 325*a^3*b*c*x^2 - 128*a^4*c)*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^3 + a*x),
-1/128*(315*(a^4*b*c*x^3 + a^5*c*x)*sqrt(-b*c)*arctan(sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*
sqrt(-b*c)*x/(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)) - (16*b^4*c*x^8 + 88*a*b^3*c*x^6 + 210*a^2*b^2*c*x^4 + 325*a^3
*b*c*x^2 - 128*a^4*c)*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^3 + a*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.64, size = 185, normalized size = 0.89 \begin {gather*} \frac {1}{256} \, {\left (\frac {512 \, \sqrt {b c} a^{5} c \mathrm {sgn}\left (b x^{2} + a\right )}{{\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c}\right )}^{2} - a c} - 315 \, \sqrt {b c} a^{4} \log \left ({\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c}\right )}^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (325 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (105 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (2 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 11 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\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((c*(b*x^2+a)^3)^(3/2)/x^2,x, algorithm="giac")

[Out]

1/256*(512*sqrt(b*c)*a^5*c*sgn(b*x^2 + a)/((sqrt(b*c)*x - sqrt(b*c*x^2 + a*c))^2 - a*c) - 315*sqrt(b*c)*a^4*lo
g((sqrt(b*c)*x - sqrt(b*c*x^2 + a*c))^2)*sgn(b*x^2 + a) + 2*(325*a^3*b*sgn(b*x^2 + a) + 2*(105*a^2*b^2*sgn(b*x
^2 + a) + 4*(2*b^4*x^2*sgn(b*x^2 + a) + 11*a*b^3*sgn(b*x^2 + a))*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 \frac {{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(a + b*x^2)^3)^(3/2)/x^2,x)

[Out]

int((c*(a + b*x^2)^3)^(3/2)/x^2, x)

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