3.4.35 \(\int \frac {x^3}{(d+e x) (a+c x^2)^{3/2}} \, dx\) [335]

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

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 = {1661, 858, 223, 212, 739} \begin {gather*} \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}} \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 739

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]

Rule 858

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 1661

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]

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 {\text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c e}+\frac {d^3 \text {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*}

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Mathematica [A]
time = 0.57, size = 137, normalized size = 1.11 \begin {gather*} \frac {a (d-e x)}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {2 d^3 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{e \left (-c d^2-a e^2\right )^{3/2}}-\frac {\log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2} e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[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]) - (2*d^3*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt
[-(c*d^2) - a*e^2]])/(e*(-(c*d^2) - a*e^2)^(3/2)) - Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]]/(c^(3/2)*e)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(396\) vs. \(2(109)=218\).
time = 0.07, size = 397, normalized size = 3.23

method	result	size
default	\(\frac {-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}}{e}+\frac {d}{e^{2} c \sqrt {c \,x^{2}+a}}+\frac {d^{2} x}{e^{3} a \sqrt {c \,x^{2}+a}}-\frac {d^{3} \left (\frac {e^{2}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}+c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\)	\(397\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 204, normalized size = 1.66 \begin {gather*} -\frac {c d^{4} x}{\sqrt {c x^{2} + a} a c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{2} e^{5}} - \frac {d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-4\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} + \frac {d^{2} x e^{\left (-3\right )}}{\sqrt {c x^{2} + a} a} - \frac {d^{3}}{\sqrt {c x^{2} + a} c d^{2} e^{2} + \sqrt {c x^{2} + a} a e^{4}} - \frac {x e^{\left (-1\right )}}{\sqrt {c x^{2} + a} c} + \frac {\operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-1\right )}}{c^{\frac {3}{2}}} + \frac {d e^{\left (-2\right )}}{\sqrt {c x^{2} + a} c} \end {gather*}

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]

-c*d^4*x/(sqrt(c*x^2 + a)*a*c*d^2*e^3 + sqrt(c*x^2 + a)*a^2*e^5) - d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d))
- a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-4)/(c*d^2*e^(-2) + a)^(3/2) + d^2*x*e^(-3)/(sqrt(c*x^2 + a)*a) - d^3/(sqrt
(c*x^2 + a)*c*d^2*e^2 + sqrt(c*x^2 + a)*a*e^4) - x*e^(-1)/(sqrt(c*x^2 + a)*c) + arcsinh(c*x/sqrt(a*c))*e^(-1)/
c^(3/2) + d*e^(-2)/(sqrt(c*x^2 + a)*c)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (109) = 218\).
time = 9.96, size = 1304, normalized size = 10.60 \begin {gather*} \left [\frac {{\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (a c^{2} d^{2} x e^{2} - a c^{2} d^{3} e + a^{2} c x e^{4} - a^{2} c d e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left ({\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} e^{5} + 2 \, {\left (a c^{4} d^{2} x^{2} + a^{2} c^{3} d^{2}\right )} e^{3} + {\left (c^{5} d^{4} x^{2} + a c^{4} d^{4}\right )} e\right )}}, -\frac {2 \, {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) - {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (a c^{2} d^{2} x e^{2} - a c^{2} d^{3} e + a^{2} c x e^{4} - a^{2} c d e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left ({\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} e^{5} + 2 \, {\left (a c^{4} d^{2} x^{2} + a^{2} c^{3} d^{2}\right )} e^{3} + {\left (c^{5} d^{4} x^{2} + a c^{4} d^{4}\right )} e\right )}}, -\frac {2 \, {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (a c^{2} d^{2} x e^{2} - a c^{2} d^{3} e + a^{2} c x e^{4} - a^{2} c d e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left ({\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} e^{5} + 2 \, {\left (a c^{4} d^{2} x^{2} + a^{2} c^{3} d^{2}\right )} e^{3} + {\left (c^{5} d^{4} x^{2} + a c^{4} d^{4}\right )} e\right )}}, -\frac {{\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} + {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 2 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (a c^{2} d^{2} x e^{2} - a c^{2} d^{3} e + a^{2} c x e^{4} - a^{2} c d e^{3}\right )} \sqrt {c x^{2} + a}}{{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )} e^{5} + 2 \, {\left (a c^{4} d^{2} x^{2} + a^{2} c^{3} d^{2}\right )} e^{3} + {\left (c^{5} d^{4} x^{2} + a c^{4} d^{4}\right )} e}\right ] \end {gather*}

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

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

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (109) = 218\).
time = 0.86, size = 219, normalized size = 1.78 \begin {gather*} -\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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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