3.4.45 \(\int \frac {x^2}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [345]

Optimal. Leaf size=137 \[ -\frac {d^2 \sqrt {a+c x^2}}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}+\frac {d \left (c d^2+2 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 137, 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 = {1665, 858, 223, 212, 739} \begin {gather*} -\frac {d^2 \sqrt {a+c x^2}}{e (d+e x) \left (a e^2+c d^2\right )}+\frac {d \left (2 a e^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^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.69, size = 146, normalized size = 1.07 \begin {gather*} -\frac {\frac {d^2 e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {2 d \left (c d^2+2 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 {\log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(368\) vs. \(2(123)=246\).
time = 0.07, size = 369, normalized size = 2.69

method	result	size
default	\(\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{2} \sqrt {c}}+\frac {2 d \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {d^{2} \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\)	\(369\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^2*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)+2*d/e^3/((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))+1/e^4*d^2*
(-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 = 170, normalized size = 1.24 \begin {gather*} \frac {c 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 (-5\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} - \frac {2 \, d \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 (-3\right )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {\sqrt {c x^{2} + a} d^{2}}{c d^{2} x e^{2} + c d^{3} e + a x e^{4} + a d e^{3}} + \frac {\operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (119) = 238\).
time = 10.63, size = 1208, normalized size = 8.82 \begin {gather*} \left [\frac {{\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d 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^{3} x e + c^{2} d^{4} + 2 \, a c d x e^{3} + 2 \, a c d^{2} 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} e + a c d^{2} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{4} x e^{3} + c^{3} d^{5} e^{2} + 2 \, a c^{2} d^{2} x e^{5} + 2 \, a c^{2} d^{3} e^{4} + a^{2} c x e^{7} + a^{2} c d e^{6}\right )}}, -\frac {2 \, {\left (c^{2} d^{3} x e + c^{2} d^{4} + 2 \, a c d x e^{3} + 2 \, a c d^{2} 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^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (c^{2} d^{4} e + a c d^{2} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{4} x e^{3} + c^{3} d^{5} e^{2} + 2 \, a c^{2} d^{2} x e^{5} + 2 \, a c^{2} d^{3} e^{4} + a^{2} c x e^{7} + a^{2} c d e^{6}\right )}}, -\frac {2 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (c^{2} d^{3} x e + c^{2} d^{4} + 2 \, a c d x e^{3} + 2 \, a c d^{2} 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} e + a c d^{2} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{4} x e^{3} + c^{3} d^{5} e^{2} + 2 \, a c^{2} d^{2} x e^{5} + 2 \, a c^{2} d^{3} e^{4} + a^{2} c x e^{7} + a^{2} c d e^{6}\right )}}, -\frac {{\left (c^{2} d^{3} x e + c^{2} d^{4} + 2 \, a c d x e^{3} + 2 \, a c d^{2} 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^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (c^{2} d^{4} e + a c d^{2} e^{3}\right )} \sqrt {c x^{2} + a}}{c^{3} d^{4} x e^{3} + c^{3} d^{5} e^{2} + 2 \, a c^{2} d^{2} x e^{5} + 2 \, a c^{2} d^{3} e^{4} + a^{2} c x e^{7} + a^{2} c d e^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\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**2/(e*x+d)**2/(c*x**2+a)**(1/2),x)

[Out]

Integral(x**2/(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^2/(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^2}{\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^2/((a + c*x^2)^(1/2)*(d + e*x)^2),x)

[Out]

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

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