3.335 \(\int \frac{x^3}{(d+e x) (a+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e}+\frac{a (d-e x)}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}} \]

[Out]

(a*(d - e*x))/(c*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]/(c^(3/2)*e) + (d^3*Ar
cTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e*(c*d^2 + a*e^2)^(3/2))

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Rubi [A]  time = 0.164837, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1647, 844, 217, 206, 725} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e}+\frac{a (d-e x)}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

(a*(d - e*x))/(c*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]/(c^(3/2)*e) + (d^3*Ar
cTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e*(c*d^2 + a*e^2)^(3/2))

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 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 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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{x^3}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\frac{a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}-\frac{\int \frac{-\frac{a^2 d e}{c d^2+a e^2}-a x}{(d+e x) \sqrt{a+c x^2}} \, dx}{a c}\\ &=\frac{a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{\int \frac{1}{\sqrt{a+c x^2}} \, dx}{c e}-\frac{d^3 \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{e \left (c d^2+a e^2\right )}\\ &=\frac{a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{c e}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{e \left (c d^2+a e^2\right )}\\ &=\frac{a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{e \left (c d^2+a e^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.317333, size = 153, normalized size = 1.24 \[ \frac{\frac{\sqrt{c} \left (c d^3 \sqrt{a+c x^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )+a e (d-e x) \sqrt{a e^2+c d^2}\right )}{\left (a e^2+c d^2\right )^{3/2}}+\sqrt{a} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2} e \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] + (Sqrt[c]*(a*e*Sqrt[c*d^2 + a*e^2]*(d - e*x) + c*d^
3*Sqrt[a + c*x^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]))/(c*d^2 + a*e^2)^(3/2))/(c^(3/
2)*e*Sqrt[a + c*x^2])

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Maple [B]  time = 0.242, size = 354, normalized size = 2.9 \begin{align*} -{\frac{x}{ce}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{1}{e}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{d}{c{e}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{d}^{2}x}{a{e}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{{d}^{3}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{4}xc}{{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{d}^{3}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(e*x+d)/(c*x^2+a)^(3/2),x)

[Out]

-1/e*x/c/(c*x^2+a)^(1/2)+1/e/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+d/e^2/c/(c*x^2+a)^(1/2)+d^2/e^3*x/a/(c*x^2+
a)^(1/2)-d^3/e^2/(a*e^2+c*d^2)/((d/e+x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-d^4/e^3/(a*e^2+c*d^2)/a/(
(d/e+x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x*c+d^3/e^2/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2
*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(1/2))/(d/e+x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 32.7009, size = 2691, normalized size = 21.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*x^2)*sqrt(c)*log(-2*c*x
^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (c^3*d^3*x^2 + a*c^2*d^3)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d
^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2
*d*e*x + d^2)) + 2*(a*c^2*d^3*e + a^2*c*d*e^3 - (a*c^2*d^2*e^2 + a^2*c*e^4)*x)*sqrt(c*x^2 + a))/(a*c^4*d^4*e +
2*a^2*c^3*d^2*e^3 + a^3*c^2*e^5 + (c^5*d^4*e + 2*a*c^4*d^2*e^3 + a^2*c^3*e^5)*x^2), 1/2*(2*(c^3*d^3*x^2 + a*c
^2*d^3)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c
^2*d^2 + a*c*e^2)*x^2)) + (a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*x^2
)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(a*c^2*d^3*e + a^2*c*d*e^3 - (a*c^2*d^2*e^2 + a^
2*c*e^4)*x)*sqrt(c*x^2 + a))/(a*c^4*d^4*e + 2*a^2*c^3*d^2*e^3 + a^3*c^2*e^5 + (c^5*d^4*e + 2*a*c^4*d^2*e^3 + a
^2*c^3*e^5)*x^2), -1/2*(2*(a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*x^2
)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (c^3*d^3*x^2 + a*c^2*d^3)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x
- a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2
*x^2 + 2*d*e*x + d^2)) - 2*(a*c^2*d^3*e + a^2*c*d*e^3 - (a*c^2*d^2*e^2 + a^2*c*e^4)*x)*sqrt(c*x^2 + a))/(a*c^4
*d^4*e + 2*a^2*c^3*d^2*e^3 + a^3*c^2*e^5 + (c^5*d^4*e + 2*a*c^4*d^2*e^3 + a^2*c^3*e^5)*x^2), ((c^3*d^3*x^2 + a
*c^2*d^3)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 +
(c^2*d^2 + a*c*e^2)*x^2)) - (a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4 + (c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*x
^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (a*c^2*d^3*e + a^2*c*d*e^3 - (a*c^2*d^2*e^2 + a^2*c*e^4)*x)*
sqrt(c*x^2 + a))/(a*c^4*d^4*e + 2*a^2*c^3*d^2*e^3 + a^3*c^2*e^5 + (c^5*d^4*e + 2*a*c^4*d^2*e^3 + a^2*c^3*e^5)*
x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19655, size = 296, normalized size = 2.41 \begin{align*} -\frac{2 \, d^{3} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} e + a e^{3}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{\frac{{\left (a c^{2} d^{2} e^{3} + a^{2} c e^{5}\right )} x}{c^{4} d^{4} e^{2} + 2 \, a c^{3} d^{2} e^{4} + a^{2} c^{2} e^{6}} - \frac{a c^{2} d^{3} e^{2} + a^{2} c d e^{4}}{c^{4} d^{4} e^{2} + 2 \, a c^{3} d^{2} e^{4} + a^{2} c^{2} e^{6}}}{\sqrt{c x^{2} + a}} - \frac{e^{\left (-1\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c*d^2*e + a*e^3)*sqrt(-c*
d^2 - a*e^2)) - ((a*c^2*d^2*e^3 + a^2*c*e^5)*x/(c^4*d^4*e^2 + 2*a*c^3*d^2*e^4 + a^2*c^2*e^6) - (a*c^2*d^3*e^2
+ a^2*c*d*e^4)/(c^4*d^4*e^2 + 2*a*c^3*d^2*e^4 + a^2*c^2*e^6))/sqrt(c*x^2 + a) - e^(-1)*log(abs(-sqrt(c)*x + sq
rt(c*x^2 + a)))/c^(3/2)