3.4.42 \(\int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [342]

Optimal. Leaf size=244 \[ \frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}-\frac {d^4 \left (4 c d^2+5 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

-d*(-a*e^2+4*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^5-d^4*(5*a*e^2+4*c*d^2)*arctanh((-c*d*x+a*e)/
(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^5/(a*e^2+c*d^2)^(3/2)+1/3*(-2*a*e^2+13*c*d^2)*(c*x^2+a)^(1/2)/c^2/e^4+d
^5*(c*x^2+a)^(1/2)/e^4/(a*e^2+c*d^2)/(e*x+d)-5/3*d*(e*x+d)*(c*x^2+a)^(1/2)/c/e^4+1/3*(e*x+d)^2*(c*x^2+a)^(1/2)
/c/e^4

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Rubi [A]
time = 0.56, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, 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 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}+\frac {\sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right )}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac {d^4 \left (5 a e^2+4 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^5 \left (a e^2+c d^2\right )^{3/2}}-\frac {5 d \sqrt {a+c x^2} (d+e x)}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((d + e*x)^2*Sqrt[a + c*x^2]),x]

[Out]

((13*c*d^2 - 2*a*e^2)*Sqrt[a + c*x^2])/(3*c^2*e^4) + (d^5*Sqrt[a + c*x^2])/(e^4*(c*d^2 + a*e^2)*(d + e*x)) - (
5*d*(d + e*x)*Sqrt[a + c*x^2])/(3*c*e^4) + ((d + e*x)^2*Sqrt[a + c*x^2])/(3*c*e^4) - (d*(4*c*d^2 - a*e^2)*ArcT
anh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(c^(3/2)*e^5) - (d^4*(4*c*d^2 + 5*a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 +
a*e^2]*Sqrt[a + c*x^2])])/(e^5*(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^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx &=\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {-\frac {a d^4}{e^3}+\frac {d^3 \left (c d^2+a e^2\right ) x}{e^4}-\frac {d^2 \left (c d^2+a e^2\right ) x^2}{e^3}+d \left (a+\frac {c d^2}{e^2}\right ) x^3-\frac {\left (c d^2+a e^2\right ) x^4}{e}}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2}\\ &=\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-a d^2 e \left (c d^2-2 a e^2\right )+4 d \left (c d^2+a e^2\right )^2 x+2 e \left (c d^2+a e^2\right )^2 x^2+10 c d e^2 \left (c d^2+a e^2\right ) x^3}{(d+e x) \sqrt {a+c x^2}} \, dx}{3 c e^4 \left (c d^2+a e^2\right )}\\ &=\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-6 a c d^2 e^4 \left (2 c d^2+a e^2\right )-2 c d e^3 \left (c d^2+a e^2\right )^2 x-2 c e^4 \left (13 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^2 e^7 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-6 a c^2 d^2 e^6 \left (2 c d^2+a e^2\right )+6 c^2 d e^5 \left (4 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^3 e^9 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\left (d \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c e^5}+\frac {\left (d^4 \left (4 c d^2+5 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^5 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\left (d \left (4 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c e^5}-\frac {\left (d^4 \left (4 c d^2+5 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^5 \left (c d^2+a e^2\right )}\\ &=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}-\frac {d^4 \left (4 c d^2+5 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^5 \left (c d^2+a e^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 1.24, size = 251, normalized size = 1.03 \begin {gather*} \frac {\frac {e \sqrt {a+c x^2} \left (-2 a^2 e^4 (d+e x)+a c e^2 \left (7 d^3+4 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+c^2 d^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{c^2 \left (c d^2+a e^2\right ) (d+e x)}+\frac {6 d^4 \left (4 c d^2+5 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 {3 \left (4 c d^3-a d e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((d + e*x)^2*Sqrt[a + c*x^2]),x]

[Out]

((e*Sqrt[a + c*x^2]*(-2*a^2*e^4*(d + e*x) + a*c*e^2*(7*d^3 + 4*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + c^2*d^2*(12*
d^3 + 6*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(c^2*(c*d^2 + a*e^2)*(d + e*x)) + (6*d^4*(4*c*d^2 + 5*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) + (3*(4*c*d^3 - a*d
*e^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/c^(3/2))/(3*e^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(218)=436\).
time = 0.10, size = 477, normalized size = 1.95

method	result	size
risch	\(-\frac {\left (-c \,e^{2} x^{2}+3 c d e x +2 a \,e^{2}-9 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{3 c^{2} e^{4}}+\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a}{c^{\frac {3}{2}} e^{3}}-\frac {4 d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{5} \sqrt {c}}-\frac {5 d^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {d^{5} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{e^{5} \left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {c \,d^{6} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\)	\(442\)
default	\(\frac {\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}}{e^{2}}-\frac {2 d \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{e^{3}}+\frac {3 d^{2} \sqrt {c \,x^{2}+a}}{e^{4} c}-\frac {4 d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{5} \sqrt {c}}-\frac {5 d^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {d^{5} \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{7}}\)	\(477\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(e*x+d)^2/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/e^2*(1/3*x^2/c*(c*x^2+a)^(1/2)-2/3*a/c^2*(c*x^2+a)^(1/2))-2*d/e^3*(1/2*x/c*(c*x^2+a)^(1/2)-1/2*a/c^(3/2)*ln(
x*c^(1/2)+(c*x^2+a)^(1/2)))+3*d^2/e^4/c*(c*x^2+a)^(1/2)-4*d^3/e^5*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-5/e^6*
d^4/((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^5/e^7*(-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 = 0.33, size = 266, normalized size = 1.09 \begin {gather*} -\frac {c d^{6} \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 (-8\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} + \frac {5 \, 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 )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} + \frac {\sqrt {c x^{2} + a} d^{5}}{c d^{2} x e^{5} + c d^{3} e^{4} + a x e^{7} + a d e^{6}} - \frac {4 \, d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-5\right )}}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} x^{2} e^{\left (-2\right )}}{3 \, c} - \frac {\sqrt {c x^{2} + a} d x e^{\left (-3\right )}}{c} + \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-3\right )}}{c^{\frac {3}{2}}} + \frac {3 \, \sqrt {c x^{2} + a} d^{2} e^{\left (-4\right )}}{c} - \frac {2 \, \sqrt {c x^{2} + a} a e^{\left (-2\right )}}{3 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

-c*d^6*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-8)/(c*d^2*e^(-2) + a)^(3/2)
+ 5*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-6)/sqrt(c*d^2*e^(-2) + a) +
sqrt(c*x^2 + a)*d^5/(c*d^2*x*e^5 + c*d^3*e^4 + a*x*e^7 + a*d*e^6) - 4*d^3*arcsinh(c*x/sqrt(a*c))*e^(-5)/sqrt(
c) + 1/3*sqrt(c*x^2 + a)*x^2*e^(-2)/c - sqrt(c*x^2 + a)*d*x*e^(-3)/c + a*d*arcsinh(c*x/sqrt(a*c))*e^(-3)/c^(3/
2) + 3*sqrt(c*x^2 + a)*d^2*e^(-4)/c - 2/3*sqrt(c*x^2 + a)*a*e^(-2)/c^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (211) = 422\).
time = 69.12, size = 1960, normalized size = 8.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/6*(3*(4*c^3*d^7*x*e + 4*c^3*d^8 + 7*a*c^2*d^5*x*e^3 + 7*a*c^2*d^6*e^2 + 2*a^2*c*d^3*x*e^5 + 2*a^2*c*d^4*e^4
- a^3*d*x*e^7 - a^3*d^2*e^6)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*(4*c^3*d^6*x*e + 4*c
^3*d^7 + 5*a*c^2*d^4*x*e^3 + 5*a*c^2*d^5*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*
(6*c^3*d^6*x*e^2 + 12*c^3*d^7*e + (a^2*c*x^3 - 2*a^3*x)*e^8 - 2*(a^2*c*d*x^2 + a^3*d)*e^7 + 2*(a*c^2*d^2*x^3 +
a^2*c*d^2*x)*e^6 - (4*a*c^2*d^3*x^2 - 5*a^2*c*d^3)*e^5 + (c^3*d^4*x^3 + 10*a*c^2*d^4*x)*e^4 - (2*c^3*d^5*x^2
- 19*a*c^2*d^5)*e^3)*sqrt(c*x^2 + a))/(c^4*d^4*x*e^6 + c^4*d^5*e^5 + 2*a*c^3*d^2*x*e^8 + 2*a*c^3*d^3*e^7 + a^2
*c^2*x*e^10 + a^2*c^2*d*e^9), 1/6*(6*(4*c^3*d^6*x*e + 4*c^3*d^7 + 5*a*c^2*d^4*x*e^3 + 5*a*c^2*d^5*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)) + 3*(4*c^3*d^7*x*e + 4*c^3*d^8 + 7*a*c^2*d^5*x*e^3 + 7*a*c^2*d^6*e^2 + 2*a^2*c*d^3*x*e^5 + 2*a^2*c*d
^4*e^4 - a^3*d*x*e^7 - a^3*d^2*e^6)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(6*c^3*d^6*x*e
^2 + 12*c^3*d^7*e + (a^2*c*x^3 - 2*a^3*x)*e^8 - 2*(a^2*c*d*x^2 + a^3*d)*e^7 + 2*(a*c^2*d^2*x^3 + a^2*c*d^2*x)*
e^6 - (4*a*c^2*d^3*x^2 - 5*a^2*c*d^3)*e^5 + (c^3*d^4*x^3 + 10*a*c^2*d^4*x)*e^4 - (2*c^3*d^5*x^2 - 19*a*c^2*d^5
)*e^3)*sqrt(c*x^2 + a))/(c^4*d^4*x*e^6 + c^4*d^5*e^5 + 2*a*c^3*d^2*x*e^8 + 2*a*c^3*d^3*e^7 + a^2*c^2*x*e^10 +
a^2*c^2*d*e^9), 1/6*(6*(4*c^3*d^7*x*e + 4*c^3*d^8 + 7*a*c^2*d^5*x*e^3 + 7*a*c^2*d^6*e^2 + 2*a^2*c*d^3*x*e^5 +
2*a^2*c*d^4*e^4 - a^3*d*x*e^7 - a^3*d^2*e^6)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 3*(4*c^3*d^6*x*e +
4*c^3*d^7 + 5*a*c^2*d^4*x*e^3 + 5*a*c^2*d^5*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*(6*c^3*d^6*x*e^2 + 12*c^3*d^7*e + (a^2*c*x^3 - 2*a^3*x)*e^8 - 2*(a^2*c*d*x^2 + a^3*d)*e^7 + 2*(a*c^2*d^2*x^
3 + a^2*c*d^2*x)*e^6 - (4*a*c^2*d^3*x^2 - 5*a^2*c*d^3)*e^5 + (c^3*d^4*x^3 + 10*a*c^2*d^4*x)*e^4 - (2*c^3*d^5*x
^2 - 19*a*c^2*d^5)*e^3)*sqrt(c*x^2 + a))/(c^4*d^4*x*e^6 + c^4*d^5*e^5 + 2*a*c^3*d^2*x*e^8 + 2*a*c^3*d^3*e^7 +
a^2*c^2*x*e^10 + a^2*c^2*d*e^9), 1/3*(3*(4*c^3*d^6*x*e + 4*c^3*d^7 + 5*a*c^2*d^4*x*e^3 + 5*a*c^2*d^5*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)) + 3*(4*c^3*d^7*x*e + 4*c^3*d^8 + 7*a*c^2*d^5*x*e^3 + 7*a*c^2*d^6*e^2 + 2*a^2*c*d^3*x*e^5 + 2*a^2*
c*d^4*e^4 - a^3*d*x*e^7 - a^3*d^2*e^6)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (6*c^3*d^6*x*e^2 + 12*c^3
*d^7*e + (a^2*c*x^3 - 2*a^3*x)*e^8 - 2*(a^2*c*d*x^2 + a^3*d)*e^7 + 2*(a*c^2*d^2*x^3 + a^2*c*d^2*x)*e^6 - (4*a*
c^2*d^3*x^2 - 5*a^2*c*d^3)*e^5 + (c^3*d^4*x^3 + 10*a*c^2*d^4*x)*e^4 - (2*c^3*d^5*x^2 - 19*a*c^2*d^5)*e^3)*sqrt
(c*x^2 + a))/(c^4*d^4*x*e^6 + c^4*d^5*e^5 + 2*a*c^3*d^2*x*e^8 + 2*a*c^3*d^3*e^7 + a^2*c^2*x*e^10 + a^2*c^2*d*e
^9)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\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**5/(e*x+d)**2/(c*x**2+a)**(1/2),x)

[Out]

Integral(x**5/(sqrt(a + c*x**2)*(d + e*x)**2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{\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^5/((a + c*x^2)^(1/2)*(d + e*x)^2),x)

[Out]

int(x^5/((a + c*x^2)^(1/2)*(d + e*x)^2), x)

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