3.5.88 \(\int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx\)
Optimal. Leaf size=198 \[ -\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac {c e \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 198, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used =
{745, 835, 807, 725, 206} \begin {gather*} -\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac {c e \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]
[Out]
-(e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (5*c*d*e*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^2*(d + e*x
)^2) - (c*e*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^3*(d + e*x)) - (c^2*d*(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])])/(2*(c*d^2 + a*e^2)^(7/2))
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]
Rule 745
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])
Rule 807
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]
Rule 835
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(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*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx &=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {c \int \frac {-3 d+2 e x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c \int \frac {2 \left (3 c d^2-2 a e^2\right )-5 c d e x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{6 \left (c d^2+a e^2\right )^2}\\ &=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^3}\\ &=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \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 )}{2 \left (c d^2+a e^2\right )^3}\\ &=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {c^2 d \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 )}{2 \left (c d^2+a e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 209, normalized size = 1.06 \begin {gather*} \frac {-3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )+3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log (d+e x)-e \sqrt {a+c x^2} \sqrt {a e^2+c d^2} \left (5 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (11 c d^2-4 a e^2\right )+2 \left (a e^2+c d^2\right )^2\right )}{6 (d+e x)^3 \left (a e^2+c d^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]
[Out]
(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(2*(c*d^2 + a*e^2)^2 + 5*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2
- 4*a*e^2)*(d + e*x)^2)) + 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[d + e*x] - 3*c^2*d*(2*c*d^2 - 3*a*e^2)*
(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^3)
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IntegrateAlgebraic [B] time = 14.38, size = 4682, normalized size = 23.65 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]
[Out]
(2*a^16*e^5 - 54*a^15*Sqrt[c]*e^5*x*Sqrt[a + c*x^2] + c*(5*a^15*d^2*e^3 - 3*a^15*d*e^4*x + 726*a^15*e^5*x^2) +
c^(3/2)*Sqrt[a + c*x^2]*(-4*a^14*d^3*e^2 - 147*a^14*d^2*e^3*x + 69*a^14*d*e^4*x^2 - 6448*a^14*e^5*x^3) + c^2*
(18*a^14*d^4*e + 135*a^14*d^3*e^2*x + 2160*a^14*d^2*e^3*x^2 - 771*a^14*d*e^4*x^3 + 43056*a^14*e^5*x^4) + c^(5/
2)*Sqrt[a + c*x^2]*(11*a^13*d^5 - 453*a^13*d^4*e*x - 2152*a^13*d^3*e^2*x^2 - 21034*a^13*d^2*e^3*x^3 + 5460*a^1
3*d*e^4*x^4 - 224224*a^13*e^5*x^5) + c^3*(-297*a^13*d^5*x + 5679*a^13*d^4*e*x^2 + 22068*a^13*d^3*e^2*x^3 + 154
050*a^13*d^2*e^3*x^4 - 27300*a^13*d*e^4*x^5 + 992992*a^13*e^5*x^6) + c^(7/2)*Sqrt[a + c*x^2]*(4004*a^12*d^5*x^
2 - 46956*a^12*d^4*e*x^3 - 163800*a^12*d^3*e^2*x^4 - 883792*a^12*d^2*e^3*x^5 + 91728*a^12*d*e^4*x^6 - 3569280*
a^12*e^5*x^7) + c^4*(-36036*a^12*d^5*x^3 + 291564*a^12*d^4*e*x^4 + 963144*a^12*d^3*e^2*x^5 + 4295200*a^12*d^2*
e^3*x^6 - 187824*a^12*d*e^4*x^7 + 11806080*a^12*e^5*x^8) + c^(9/2)*Sqrt[a + c*x^2]*(240240*a^11*d^5*x^4 - 1402
128*a^11*d^4*e*x^5 - 4513600*a^11*d^3*e^2*x^6 - 17091360*a^11*d^2*e^3*x^7 - 219648*a^11*d*e^4*x^8 - 31482880*a
^11*e^5*x^9) + c^5*(-1297296*a^11*d^5*x^5 + 5726448*a^11*d^4*e*x^6 + 18442944*a^11*d^3*e^2*x^7 + 61981920*a^11
*d^2*e^3*x^8 + 3514368*a^11*d*e^4*x^9 + 83026944*a^11*e^5*x^10) + c^(11/2)*Sqrt[a + c*x^2]*(5637632*a^10*d^5*x
^6 - 18670080*a^10*d^4*e*x^7 - 61355008*a^10*d^3*e^2*x^8 - 184138240*a^10*d^2*e^3*x^9 - 18670080*a^10*d*e^4*x^
10 - 170180608*a^10*e^5*x^11) + c^6*(-21745152*a^10*d^5*x^7 + 56010240*a^10*d^4*e*x^8 + 191313408*a^10*d^3*e^2
*x^9 + 531548160*a^10*d^2*e^3*x^10 + 70946304*a^10*d*e^4*x^11 + 372948992*a^10*e^5*x^12) + c^(13/2)*Sqrt[a + c
*x^2]*(68456960*a^9*d^5*x^8 - 130690560*a^9*d^4*e*x^9 - 477954048*a^9*d^3*e^2*x^10 - 1224530944*a^9*d^2*e^3*x^
11 - 207747072*a^9*d*e^4*x^12 - 596606976*a^9*e^5*x^13) + c^7*(-205370880*a^9*d^5*x^9 + 302455296*a^9*d^4*e*x^
10 + 1201674240*a^9*d^3*e^2*x^11 + 2931768320*a^9*d^2*e^3*x^12 + 566212608*a^9*d*e^4*x^13 + 1118011392*a^9*e^5
*x^14) + c^(15/2)*Sqrt[a + c*x^2]*(492890112*a^8*d^5*x^10 - 501037056*a^8*d^4*e*x^11 - 2316447744*a^8*d^3*e^2*
x^12 - 5336875008*a^8*d^2*e^3*x^13 - 1190707200*a^8*d*e^4*x^14 - 1397096448*a^8*e^5*x^15) + c^8*(-1209821184*a
^8*d^5*x^11 + 892090368*a^8*d^4*e*x^12 + 4867485696*a^8*d^3*e^2*x^13 + 10904371200*a^8*d^2*e^3*x^14 + 26195558
40*a^8*d*e^4*x^15 + 2286157824*a^8*e^5*x^16) + c^(17/2)*Sqrt[a + c*x^2]*(2270281728*a^7*d^5*x^12 - 904937472*a
^7*d^4*e*x^13 - 7303004160*a^7*d^3*e^2*x^14 - 15780839424*a^7*d^2*e^3*x^15 - 4191289344*a^7*d*e^4*x^16 - 22039
75680*a^7*e^5*x^17) + c^9*(-4715200512*a^7*d^5*x^13 + 1000194048*a^7*d^4*e*x^14 + 13145407488*a^7*d^3*e^2*x^15
+ 28100689920*a^7*d^2*e^3*x^16 + 7811039232*a^7*d*e^4*x^17 + 3200122880*a^7*e^5*x^18) + c^(19/2)*Sqrt[a + c*x
^2]*(6985482240*a^6*d^5*x^14 + 381026304*a^6*d^4*e*x^15 - 15241052160*a^6*d^3*e^2*x^16 - 32181780480*a^6*d^2*e
^3*x^17 - 9648930816*a^6*d*e^4*x^18 - 2312110080*a^6*e^5*x^19) + c^10*(-12573868032*a^6*d^5*x^15 - 2286157824*
a^6*d^4*e*x^16 + 23903797248*a^6*d^3*e^2*x^17 + 50753699840*a^6*d^2*e^3*x^18 + 15663169536*a^6*d*e^4*x^19 + 30
16753152*a^6*e^5*x^20) + c^(21/2)*Sqrt[a + c*x^2]*(14669512704*a^5*d^5*x^16 + 5883494400*a^5*d^4*e*x^17 - 2084
9360896*a^5*d^3*e^2*x^18 - 45265059840*a^5*d^2*e^3*x^19 - 14836039680*a^5*d*e^4*x^20 - 1545601024*a^5*e^5*x^21
) + c^11*(-23298637824*a^5*d^5*x^17 - 11374755840*a^5*d^4*e*x^18 + 28785770496*a^5*d^3*e^2*x^19 + 64037191680*
a^5*d^2*e^3*x^20 + 21376008192*a^5*d*e^4*x^21 + 1830813696*a^5*e^5*x^22) + c^(23/2)*Sqrt[a + c*x^2]*(210934169
60*a^4*d^5*x^18 + 14230487040*a^4*d^4*e*x^19 - 17722507264*a^4*d^3*e^2*x^20 - 43103813632*a^4*d^2*e^3*x^21 - 1
5121514496*a^4*d*e^4*x^22 - 595591168*a^4*e^5*x^23) + c^12*(-29974855680*a^4*d^5*x^19 - 21981560832*a^4*d^4*e*
x^20 + 21540372480*a^4*d^3*e^2*x^21 + 55270440960*a^4*d^2*e^3*x^22 + 19607322624*a^4*d*e^4*x^23 + 645922816*a^
4*e^5*x^24) + c^(25/2)*Sqrt[a + c*x^2]*(20427309056*a^3*d^5*x^20 + 18304991232*a^3*d^4*e*x^21 - 7813988352*a^3
*d^3*e^2*x^22 - 26522681344*a^3*d^2*e^3*x^23 - 9814671360*a^3*d*e^4*x^24 - 100663296*a^3*e^5*x^25) + c^(27/2)*
Sqrt[a + c*x^2]*(12733906944*a^2*d^5*x^22 + 13740539904*a^2*d^4*e*x^23 - 167772160*a^2*d^3*e^2*x^24 - 95126814
72*a^2*d^2*e^3*x^25 - 3674210304*a^2*d*e^4*x^26) + c^13*(-26263683072*a^3*d^5*x^21 - 24527241216*a^3*d^4*e*x^2
2 + 8040480768*a^3*d^3*e^2*x^23 + 31090278400*a^3*d^2*e^3*x^24 + 11576279040*a^3*d*e^4*x^25 + 100663296*a^3*e^
5*x^26) + c^14*(-14948499456*a^2*d^5*x^23 - 16458448896*a^2*d^4*e*x^24 - 452984832*a^2*d^3*e^2*x^25 + 10267656
192*a^2*d^2*e^3*x^26 + 3976200192*a^2*d*e^4*x^27) + c^(31/2)*Sqrt[a + c*x^2]*(738197504*d^5*x^26 + 1006632960*
d^4*e*x^27 + 402653184*d^3*e^2*x^28) + c^(29/2)*Sqrt[a + c*x^2]*(4613734400*a*d^5*x^24 + 5687476224*a*d^4*e*x^
25 + 1342177280*a*d^3*e^2*x^26 - 1509949440*a*d^2*e^3*x^27 - 603979776*a*d*e^4*x^28) + c^16*(-738197504*d^5*x^
27 - 1006632960*d^4*e*x^28 - 402653184*d^3*e^2*x^29) + c^15*(-4982833152*a*d^5*x^25 - 6190792704*a*d^4*e*x^26
- 1543503872*a*d^3*e^2*x^27 + 1509949440*a*d^2*e^3*x^28 + 603979776*a*d*e^4*x^29))/(162*a^16*Sqrt[c]*e^6*x*(d
+ e*x)^3 + 402653184*c^(33/2)*d^6*x^27*(d + e*x)^3 - 6*a^16*e^6*(d + e*x)^3*Sqrt[a + c*x^2] - 402653184*c^16*d
^6*x^26*(d + e*x)^3*Sqrt[a + c*x^2] + 6*c*(d + e*x)^3*Sqrt[a + c*x^2]*(-3*a^15*d^2*e^4 - 364*a^15*e^6*x^2) + 6
*c^(3/2)*(d + e*x)^3*(81*a^15*d^2*e^4*x + 3276*a^15*e^6*x^3) + 6*c^2*(d + e*x)^3*Sqrt[a + c*x^2]*(-3*a^14*d^4*
e^2 - 1092*a^14*d^2*e^4*x^2 - 21840*a^14*e^6*x^4) + 6*c^(5/2)*(d + e*x)^3*(81*a^14*d^4*e^2*x + 9828*a^14*d^2*e
^4*x^3 + 117936*a^14*e^6*x^5) + 6*c^3*(d + e*x)^3*Sqrt[a + c*x^2]*(-(a^13*d^6) - 1092*a^13*d^4*e^2*x^2 - 65520
*a^13*d^2*e^4*x^4 - 512512*a^13*e^6*x^6) + 6*c^(7/2)*(d + e*x)^3*(27*a^13*d^6*x + 9828*a^13*d^4*e^2*x^3 + 3538
08*a^13*d^2*e^4*x^5 + 1976832*a^13*e^6*x^7) + 6*c^4*(d + e*x)^3*Sqrt[a + c*x^2]*(-364*a^12*d^6*x^2 - 65520*a^1
2*d^4*e^2*x^4 - 1537536*a^12*d^2*e^4*x^6 - 6223360*a^12*e^6*x^8) + 6*c^(9/2)*(d + e*x)^3*(3276*a^12*d^6*x^3 +
353808*a^12*d^4*e^2*x^5 + 5930496*a^12*d^2*e^4*x^7 + 18670080*a^12*e^6*x^9) + 6*c^5*(d + e*x)^3*Sqrt[a + c*x^2
]*(-21840*a^11*d^6*x^4 - 1537536*a^11*d^4*e^2*x^6 - 18670080*a^11*d^2*e^4*x^8 - 44808192*a^11*e^6*x^10) + 6*c^
(11/2)*(d + e*x)^3*(117936*a^11*d^6*x^5 + 5930496*a^11*d^4*e^2*x^7 + 56010240*a^11*d^2*e^4*x^9 + 109983744*a^1
1*e^6*x^11) + 6*c^6*(d + e*x)^3*Sqrt[a + c*x^2]*(-512512*a^10*d^6*x^6 - 18670080*a^10*d^4*e^2*x^8 - 134424576*
a^10*d^2*e^4*x^10 - 206389248*a^10*e^6*x^12) + 6*c^(13/2)*(d + e*x)^3*(1976832*a^10*d^6*x^7 + 56010240*a^10*d^
4*e^2*x^9 + 329951232*a^10*d^2*e^4*x^11 + 428654592*a^10*e^6*x^13) + 6*c^7*(d + e*x)^3*Sqrt[a + c*x^2]*(-62233
60*a^9*d^6*x^8 - 134424576*a^9*d^4*e^2*x^10 - 619167744*a^9*d^2*e^4*x^12 - 635043840*a^9*e^6*x^14) + 6*c^(15/2
)*(d + e*x)^3*(18670080*a^9*d^6*x^9 + 329951232*a^9*d^4*e^2*x^11 + 1285963776*a^9*d^2*e^4*x^13 + 1143078912*a^
9*e^6*x^15) + 6*c^8*(d + e*x)^3*Sqrt[a + c*x^2]*(-44808192*a^8*d^6*x^10 - 619167744*a^8*d^4*e^2*x^12 - 1905131
520*a^8*d^2*e^4*x^14 - 1333592064*a^8*e^6*x^16) + 6*c^(17/2)*(d + e*x)^3*(109983744*a^8*d^6*x^11 + 1285963776*
a^8*d^4*e^2*x^13 + 3429236736*a^8*d^2*e^4*x^15 + 2118057984*a^8*e^6*x^17) + 6*c^9*(d + e*x)^3*Sqrt[a + c*x^2]*
(-206389248*a^7*d^6*x^12 - 1905131520*a^7*d^4*e^2*x^14 - 4000776192*a^7*d^2*e^4*x^16 - 1917583360*a^7*e^6*x^18
) + 6*c^(19/2)*(d + e*x)^3*(428654592*a^7*d^6*x^13 + 3429236736*a^7*d^4*e^2*x^15 + 6354173952*a^7*d^2*e^4*x^17
+ 2724986880*a^7*e^6*x^19) + 6*c^10*(d + e*x)^3*Sqrt[a + c*x^2]*(-635043840*a^6*d^6*x^14 - 4000776192*a^6*d^4
*e^2*x^16 - 5752750080*a^6*d^2*e^4*x^18 - 1857028096*a^6*e^6*x^20) + 6*c^(21/2)*(d + e*x)^3*(1143078912*a^6*d^
6*x^15 + 6354173952*a^6*d^4*e^2*x^17 + 8174960640*a^6*d^2*e^4*x^19 + 2387607552*a^6*e^6*x^21) + 6*c^11*(d + e*
x)^3*Sqrt[a + c*x^2]*(-1333592064*a^5*d^6*x^16 - 5752750080*a^5*d^4*e^2*x^18 - 5571084288*a^5*d^2*e^4*x^20 - 1
157627904*a^5*e^6*x^22) + 6*c^(23/2)*(d + e*x)^3*(2118057984*a^5*d^6*x^17 + 8174960640*a^5*d^4*e^2*x^19 + 7162
822656*a^5*d^2*e^4*x^21 + 1358954496*a^5*e^6*x^23) + 6*c^12*(d + e*x)^3*Sqrt[a + c*x^2]*(-1917583360*a^4*d^6*x
^18 - 5571084288*a^4*d^4*e^2*x^20 - 3472883712*a^4*d^2*e^4*x^22 - 419430400*a^4*e^6*x^24) + 6*c^(25/2)*(d + e*
x)^3*(2724986880*a^4*d^6*x^19 + 7162822656*a^4*d^4*e^2*x^21 + 4076863488*a^4*d^2*e^4*x^23 + 452984832*a^4*e^6*
x^25) + 6*c^15*(d + e*x)^3*Sqrt[a + c*x^2]*(-419430400*a*d^6*x^24 - 201326592*a*d^4*e^2*x^26) + 6*c^14*(d + e*
x)^3*Sqrt[a + c*x^2]*(-1157627904*a^2*d^6*x^22 - 1258291200*a^2*d^4*e^2*x^24 - 201326592*a^2*d^2*e^4*x^26) + 6
*c^13*(d + e*x)^3*Sqrt[a + c*x^2]*(-1857028096*a^3*d^6*x^20 - 3472883712*a^3*d^4*e^2*x^22 - 1258291200*a^3*d^2
*e^4*x^24 - 67108864*a^3*e^6*x^26) + 6*c^(31/2)*(d + e*x)^3*(452984832*a*d^6*x^25 + 201326592*a*d^4*e^2*x^27)
+ 6*c^(29/2)*(d + e*x)^3*(1358954496*a^2*d^6*x^23 + 1358954496*a^2*d^4*e^2*x^25 + 201326592*a^2*d^2*e^4*x^27)
+ 6*c^(27/2)*(d + e*x)^3*(2387607552*a^3*d^6*x^21 + 4076863488*a^3*d^4*e^2*x^23 + 1358954496*a^3*d^2*e^4*x^25
+ 67108864*a^3*e^6*x^27)) - (5*c^3*d^3*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2)
- a*e^2] - (e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(Sqrt[-(c*d^2) - a*e^2]*(c*d^2 + a*e^2)^3) + (3*c^2*d
*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) - a*e^2] - (e*Sqrt[a + c*x^2])/Sqrt[-
(c*d^2) - a*e^2]])/(Sqrt[-(c*d^2) - a*e^2]*(c*d^2 + a*e^2)^2)
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fricas [B] time = 1.02, size = 1139, normalized size = 5.75 \begin {gather*} \left [-\frac {3 \, {\left (2 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (18 \, c^{3} d^{6} e + 23 \, a c^{2} d^{4} e^{3} + 7 \, a^{2} c d^{2} e^{5} + 2 \, a^{3} e^{7} + {\left (11 \, c^{3} d^{4} e^{3} + 7 \, a c^{2} d^{2} e^{5} - 4 \, a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (9 \, c^{3} d^{5} e^{2} + 8 \, a c^{2} d^{3} e^{4} - a^{2} c d e^{6}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{4} d^{11} + 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} + 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} + 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} + 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e + 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} + 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}}, -\frac {3 \, {\left (2 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x\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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (18 \, c^{3} d^{6} e + 23 \, a c^{2} d^{4} e^{3} + 7 \, a^{2} c d^{2} e^{5} + 2 \, a^{3} e^{7} + {\left (11 \, c^{3} d^{4} e^{3} + 7 \, a c^{2} d^{2} e^{5} - 4 \, a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (9 \, c^{3} d^{5} e^{2} + 8 \, a c^{2} d^{3} e^{4} - a^{2} c d e^{6}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{4} d^{11} + 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} + 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} + 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} + 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e + 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} + 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="fricas")
[Out]
[-1/12*(3*(2*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*
e^4)*x^2 + 3*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*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
*(18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*
e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 +
6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^
3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e^6 + 4*a^3*c*d^3*e^8 + a^4*d*e
^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x), -1/6*(3*(2
*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 + 3
*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*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)) + (18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a
^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e
^6)*x)*sqrt(c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4
*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^
7*e^4 + 6*a^2*c^2*d^5*e^6 + 4*a^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^
6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x)]
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giac [B] time = 0.29, size = 578, normalized size = 2.92 \begin {gather*} \frac {1}{3} \, c^{\frac {3}{2}} {\left (\frac {3 \, {\left (2 \, c^{\frac {3}{2}} d^{3} - 3 \, a \sqrt {c} d e^{2}\right )} \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^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{2} d^{4} e + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{\frac {5}{2}} d^{5} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{\frac {3}{2}} d^{3} e^{2} - 102 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{2} d^{4} e - 82 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{\frac {3}{2}} d^{3} e^{2} - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c d^{2} e^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a \sqrt {c} d e^{4} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{\frac {3}{2}} d^{3} e^{2} + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} \sqrt {c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} \sqrt {c} d e^{4} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} e^{5} + 4 \, a^{4} e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="giac")
[Out]
1/3*c^(3/2)*(3*(2*c^(3/2)*d^3 - 3*a*sqrt(c)*d*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(
-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) - (30*(sqrt(c)
*x - sqrt(c*x^2 + a))^4*c^2*d^4*e + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^(5/2)*d^5 + 6*(sqrt(c)*x - sqrt(c*x^2
+ a))^5*c^(3/2)*d^3*e^2 - 102*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^2*d^4*e - 82*(sqrt(c)*x - sqrt(c*x^2 + a))^
3*a*c^(3/2)*d^3*e^2 - 45*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*sqr
t(c)*d*e^4 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c*d
^2*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*sqrt(c)*d*e^4 - 11*a^3*c*d^2*e^3 - 15*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^3*sqrt(c)*d*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*e^5 + 4*a^4*e^5)/((c^3*d^6 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d -
a*e)^3))
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maple [B] time = 0.06, size = 573, normalized size = 2.89 \begin {gather*} -\frac {5 c^{3} d^{3} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e}-\frac {5 \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, c^{2} d^{2}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3} \left (x +\frac {d}{e}\right )}+\frac {3 c^{2} d \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e}-\frac {5 \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, c d}{6 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{2} e}+\frac {2 \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, c}{3 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}-\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{3 \left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^4/(c*x^2+a)^(1/2),x)
[Out]
-1/3/e^2/(a*e^2+c*d^2)/(x+d/e)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-5/6/e*c*d/(a*e^2+c*d^2
)^2/(x+d/e)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-5/2*c^2*d^2/(a*e^2+c*d^2)^3/(x+d/e)*(-2*(
x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-5/2/e*c^3*d^3/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((
-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e
^2)^(1/2))/(x+d/e))+3/2/e*c^2*d/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)
/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+2/3/(a*e^2+c
*d^2)^2*c/(x+d/e)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)
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maxima [B] time = 1.83, size = 479, normalized size = 2.42 \begin {gather*} -\frac {5 \, \sqrt {c x^{2} + a} c^{2} d^{2}}{2 \, {\left (c^{3} d^{6} x + 3 \, a c^{2} d^{4} e^{2} x + 3 \, a^{2} c d^{2} e^{4} x + a^{3} e^{6} x + \frac {c^{3} d^{7}}{e} + 3 \, a c^{2} d^{5} e + 3 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )}} - \frac {5 \, \sqrt {c x^{2} + a} c d}{6 \, {\left (c^{2} d^{4} e x^{2} + 2 \, a c d^{2} e^{3} x^{2} + a^{2} e^{5} x^{2} + 2 \, c^{2} d^{5} x + 4 \, a c d^{3} e^{2} x + 2 \, a^{2} d e^{4} x + \frac {c^{2} d^{6}}{e} + 2 \, a c d^{4} e + a^{2} d^{2} e^{3}\right )}} + \frac {2 \, \sqrt {c x^{2} + a} c}{3 \, {\left (c^{2} d^{4} x + 2 \, a c d^{2} e^{2} x + a^{2} e^{4} x + \frac {c^{2} d^{5}}{e} + 2 \, a c d^{3} e + a^{2} d e^{3}\right )}} - \frac {\sqrt {c x^{2} + a}}{3 \, {\left (c d^{2} e^{2} x^{3} + a e^{4} x^{3} + 3 \, c d^{3} e x^{2} + 3 \, a d e^{3} x^{2} + 3 \, c d^{4} x + 3 \, a d^{2} e^{2} x + \frac {c d^{5}}{e} + a d^{3} e\right )}} + \frac {5 \, c^{3} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {7}{2}} e^{7}} - \frac {3 \, c^{2} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{2}} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="maxima")
[Out]
-5/2*sqrt(c*x^2 + a)*c^2*d^2/(c^3*d^6*x + 3*a*c^2*d^4*e^2*x + 3*a^2*c*d^2*e^4*x + a^3*e^6*x + c^3*d^7/e + 3*a*
c^2*d^5*e + 3*a^2*c*d^3*e^3 + a^3*d*e^5) - 5/6*sqrt(c*x^2 + a)*c*d/(c^2*d^4*e*x^2 + 2*a*c*d^2*e^3*x^2 + a^2*e^
5*x^2 + 2*c^2*d^5*x + 4*a*c*d^3*e^2*x + 2*a^2*d*e^4*x + c^2*d^6/e + 2*a*c*d^4*e + a^2*d^2*e^3) + 2/3*sqrt(c*x^
2 + a)*c/(c^2*d^4*x + 2*a*c*d^2*e^2*x + a^2*e^4*x + c^2*d^5/e + 2*a*c*d^3*e + a^2*d*e^3) - 1/3*sqrt(c*x^2 + a)
/(c*d^2*e^2*x^3 + a*e^4*x^3 + 3*c*d^3*e*x^2 + 3*a*d*e^3*x^2 + 3*c*d^4*x + 3*a*d^2*e^2*x + c*d^5/e + a*d^3*e) +
5/2*c^3*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(7/2)*e^7
) - 3/2*c^2*d*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(5/2)*e^
5)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + c*x^2)^(1/2)*(d + e*x)^4),x)
[Out]
int(1/((a + c*x^2)^(1/2)*(d + e*x)^4), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**4/(c*x**2+a)**(1/2),x)
[Out]
Integral(1/(sqrt(a + c*x**2)*(d + e*x)**4), x)
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