3.4.44 \(\int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [344]

Optimal. Leaf size=160 \[ \frac {\sqrt {a+c x^2}}{c e^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^3}-\frac {d^2 \left (2 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1665, 1668, 858, 223, 212, 739} \begin {gather*} -\frac {d^2 \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3 \left (a e^2+c d^2\right )^{3/2}}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^3}+\frac {\sqrt {a+c x^2}}{c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + c*x^2]/(c*e^2) + (d^3*Sqrt[a + c*x^2])/(e^2*(c*d^2 + a*e^2)*(d + e*x)) - (2*d*ArcTanh[(Sqrt[c]*x)/Sqr
t[a + c*x^2]])/(Sqrt[c]*e^3) - (d^2*(2*c*d^2 + 3*a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*
x^2])])/(e^3*(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 1665

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

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

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

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Mathematica [A]
time = 0.86, size = 173, normalized size = 1.08 \begin {gather*} \frac {\frac {e \sqrt {a+c x^2} \left (a e^2 (d+e x)+c d^2 (2 d+e x)\right )}{c \left (c d^2+a e^2\right ) (d+e x)}+\frac {2 d^2 \left (2 c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(389\) vs. \(2(144)=288\).
time = 0.08, size = 390, normalized size = 2.44

method	result	size
risch	\(\frac {\sqrt {c \,x^{2}+a}}{c \,e^{2}}-\frac {2 d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}-\frac {3 d^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {d^{3} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{e^{3} \left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {d^{4} c \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\)	\(386\)
default	\(\frac {\sqrt {c \,x^{2}+a}}{c \,e^{2}}-\frac {2 d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}-\frac {3 d^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {d^{3} \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{5}}\)	\(390\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(c*x^2+a)^(1/2)/c/e^2-2*d/e^3*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-3/e^4*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*(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))-d^3/e^5*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c*
d*e/(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*(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 = 0.31, size = 189, normalized size = 1.18 \begin {gather*} -\frac {c d^{4} \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 (-6\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} + \frac {3 \, d^{2} \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 )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} + \frac {\sqrt {c x^{2} + a} d^{3}}{c d^{2} x e^{3} + c d^{3} e^{2} + a x e^{5} + a d e^{4}} - \frac {2 \, d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-3\right )}}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e^{\left (-2\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

-c*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-6)/(c*d^2*e^(-2) + a)^(3/2)
+ 3*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-4)/sqrt(c*d^2*e^(-2) + a) +
sqrt(c*x^2 + a)*d^3/(c*d^2*x*e^3 + c*d^3*e^2 + a*x*e^5 + a*d*e^4) - 2*d*arcsinh(c*x/sqrt(a*c))*e^(-3)/sqrt(c)
+ sqrt(c*x^2 + a)*e^(-2)/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (139) = 278\).
time = 11.06, size = 1376, normalized size = 8.60 \begin {gather*} \left [\frac {2 \, {\left (c^{2} d^{5} x e + c^{2} d^{6} + 2 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (2 \, c^{2} d^{4} x e + 2 \, c^{2} d^{5} + 3 \, a c d^{2} x e^{3} + 3 \, a c d^{3} e^{2}\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 (c^{2} d^{4} x e^{2} + 2 \, c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 3 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{4} x e^{4} + c^{3} d^{5} e^{3} + 2 \, a c^{2} d^{2} x e^{6} + 2 \, a c^{2} d^{3} e^{5} + a^{2} c x e^{8} + a^{2} c d e^{7}\right )}}, \frac {{\left (2 \, c^{2} d^{4} x e + 2 \, c^{2} d^{5} + 3 \, a c d^{2} x e^{3} + 3 \, a c d^{3} e^{2}\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^{2} d^{5} x e + c^{2} d^{6} + 2 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (c^{2} d^{4} x e^{2} + 2 \, c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 3 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )} \sqrt {c x^{2} + a}}{c^{3} d^{4} x e^{4} + c^{3} d^{5} e^{3} + 2 \, a c^{2} d^{2} x e^{6} + 2 \, a c^{2} d^{3} e^{5} + a^{2} c x e^{8} + a^{2} c d e^{7}}, \frac {4 \, {\left (c^{2} d^{5} x e + c^{2} d^{6} + 2 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, c^{2} d^{4} x e + 2 \, c^{2} d^{5} + 3 \, a c d^{2} x e^{3} + 3 \, a c d^{3} e^{2}\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 (c^{2} d^{4} x e^{2} + 2 \, c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 3 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{4} x e^{4} + c^{3} d^{5} e^{3} + 2 \, a c^{2} d^{2} x e^{6} + 2 \, a c^{2} d^{3} e^{5} + a^{2} c x e^{8} + a^{2} c d e^{7}\right )}}, \frac {{\left (2 \, c^{2} d^{4} x e + 2 \, c^{2} d^{5} + 3 \, a c d^{2} x e^{3} + 3 \, a c d^{3} e^{2}\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 ) + 2 \, {\left (c^{2} d^{5} x e + c^{2} d^{6} + 2 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (c^{2} d^{4} x e^{2} + 2 \, c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 3 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )} \sqrt {c x^{2} + a}}{c^{3} d^{4} x e^{4} + c^{3} d^{5} e^{3} + 2 \, a c^{2} d^{2} x e^{6} + 2 \, a c^{2} d^{3} e^{5} + a^{2} c x e^{8} + a^{2} c d e^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/2*(2*(c^2*d^5*x*e + c^2*d^6 + 2*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + a^2*d*x*e^5 + a^2*d^2*e^4)*sqrt(c)*log(-2*c
*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (2*c^2*d^4*x*e + 2*c^2*d^5 + 3*a*c*d^2*x*e^3 + 3*a*c*d^3*e^2)*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*(c^2*d^4*x*e^2 + 2*c^2*d^5*e + 2*a*c*d^2*x*e^4 + 3*a
*c*d^3*e^3 + a^2*x*e^6 + a^2*d*e^5)*sqrt(c*x^2 + a))/(c^3*d^4*x*e^4 + c^3*d^5*e^3 + 2*a*c^2*d^2*x*e^6 + 2*a*c^
2*d^3*e^5 + a^2*c*x*e^8 + a^2*c*d*e^7), ((2*c^2*d^4*x*e + 2*c^2*d^5 + 3*a*c*d^2*x*e^3 + 3*a*c*d^3*e^2)*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^2*d^5*x*e + c^2*d^6 + 2*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + a^2*d*x*e^5 + a^2*d^2*e^4)*sqrt(c)*log(
-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (c^2*d^4*x*e^2 + 2*c^2*d^5*e + 2*a*c*d^2*x*e^4 + 3*a*c*d^3*e^3 +
a^2*x*e^6 + a^2*d*e^5)*sqrt(c*x^2 + a))/(c^3*d^4*x*e^4 + c^3*d^5*e^3 + 2*a*c^2*d^2*x*e^6 + 2*a*c^2*d^3*e^5 +
a^2*c*x*e^8 + a^2*c*d*e^7), 1/2*(4*(c^2*d^5*x*e + c^2*d^6 + 2*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + a^2*d*x*e^5 + a^
2*d^2*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (2*c^2*d^4*x*e + 2*c^2*d^5 + 3*a*c*d^2*x*e^3 + 3*a*c*
d^3*e^2)*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*(c^2*d^4*x*e^2 + 2*c^2*d^5*e + 2*a*c*
d^2*x*e^4 + 3*a*c*d^3*e^3 + a^2*x*e^6 + a^2*d*e^5)*sqrt(c*x^2 + a))/(c^3*d^4*x*e^4 + c^3*d^5*e^3 + 2*a*c^2*d^2
*x*e^6 + 2*a*c^2*d^3*e^5 + a^2*c*x*e^8 + a^2*c*d*e^7), ((2*c^2*d^4*x*e + 2*c^2*d^5 + 3*a*c*d^2*x*e^3 + 3*a*c*d
^3*e^2)*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)) + 2*(c^2*d^5*x*e + c^2*d^6 + 2*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + a^2*d*x*e^5 + a^2*d^2*
e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (c^2*d^4*x*e^2 + 2*c^2*d^5*e + 2*a*c*d^2*x*e^4 + 3*a*c*d^3*
e^3 + a^2*x*e^6 + a^2*d*e^5)*sqrt(c*x^2 + a))/(c^3*d^4*x*e^4 + c^3*d^5*e^3 + 2*a*c^2*d^2*x*e^6 + 2*a*c^2*d^3*e
^5 + a^2*c*x*e^8 + a^2*c*d*e^7)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(e*x+d)**2/(c*x**2+a)**(1/2),x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + c*x^2)^(1/2)*(d + e*x)^2),x)

[Out]

int(x^3/((a + c*x^2)^(1/2)*(d + e*x)^2), x)

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