3.6.56 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^8} \, dx\) [556]
Optimal. Leaf size=246 \[ -\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {5 a^3 c^4 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]
[Out]
-5/24*a*c^2*d*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^3/(e*x+d)^4-1/6*c*d*(-c*d*x+a*e)*(c*x^2+a)^(5/2)/(a*e
^2+c*d^2)^2/(e*x+d)^6-1/7*e*(c*x^2+a)^(7/2)/(a*e^2+c*d^2)/(e*x+d)^7-5/16*a^3*c^4*d*arctanh((-c*d*x+a*e)/(a*e^2
+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(9/2)-5/16*a^2*c^3*d*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^4
/(e*x+d)^2
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Rubi [A]
time = 0.09, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {745, 735, 739,
212} \begin {gather*} -\frac {5 a^3 c^4 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac {5 a^2 c^3 d \sqrt {a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac {5 a c^2 d \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac {c d \left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + c*x^2)^(5/2)/(d + e*x)^8,x]
[Out]
(-5*a^2*c^3*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(d + e*x)^2) - (5*a*c^2*d*(a*e - c*d*x)*(a
+ c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4) - (c*d*(a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)^
2*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(7*(c*d^2 + a*e^2)*(d + e*x)^7) - (5*a^3*c^4*d*ArcTanh[(a*e - c*d*x)/(S
qrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))
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 735
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 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 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*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx &=-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {(c d) \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx}{c d^2+a e^2}\\ &=-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {\left (5 a c^2 d\right ) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )^2}\\ &=-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {\left (5 a^2 c^3 d\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {\left (5 a^3 c^4 d\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {\left (5 a^3 c^4 d\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 )}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {5 a^3 c^4 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 10.44, size = 403, normalized size = 1.64 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (48 \left (c d^2+a e^2\right )^6-232 c d \left (c d^2+a e^2\right )^5 (d+e x)+8 c \left (c d^2+a e^2\right )^4 \left (55 c d^2+18 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right )^3 \left (200 c d^2+197 a e^2\right ) (d+e x)^3+2 c^2 \left (c d^2+a e^2\right )^2 \left (80 c^2 d^4+159 a c d^2 e^2+72 a^2 e^4\right ) (d+e x)^4-c^3 d \left (c d^2+a e^2\right ) \left (8 c^2 d^4+30 a c d^2 e^2+57 a^2 e^4\right ) (d+e x)^5-c^3 \left (8 c^3 d^6+38 a c^2 d^4 e^2+87 a^2 c d^2 e^4-48 a^3 e^6\right ) (d+e x)^6\right )}{336 e^5 \left (c d^2+a e^2\right )^4 (d+e x)^7}+\frac {5 a^3 c^4 d \log (d+e x)}{16 \left (c d^2+a e^2\right )^{9/2}}-\frac {5 a^3 c^4 d \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + c*x^2)^(5/2)/(d + e*x)^8,x]
[Out]
-1/336*(Sqrt[a + c*x^2]*(48*(c*d^2 + a*e^2)^6 - 232*c*d*(c*d^2 + a*e^2)^5*(d + e*x) + 8*c*(c*d^2 + a*e^2)^4*(5
5*c*d^2 + 18*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^3*(200*c*d^2 + 197*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2
+ a*e^2)^2*(80*c^2*d^4 + 159*a*c*d^2*e^2 + 72*a^2*e^4)*(d + e*x)^4 - c^3*d*(c*d^2 + a*e^2)*(8*c^2*d^4 + 30*a*
c*d^2*e^2 + 57*a^2*e^4)*(d + e*x)^5 - c^3*(8*c^3*d^6 + 38*a*c^2*d^4*e^2 + 87*a^2*c*d^2*e^4 - 48*a^3*e^6)*(d +
e*x)^6))/(e^5*(c*d^2 + a*e^2)^4*(d + e*x)^7) + (5*a^3*c^4*d*Log[d + e*x])/(16*(c*d^2 + a*e^2)^(9/2)) - (5*a^3*
c^4*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(16*(c*d^2 + a*e^2)^(9/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15955\) vs.
\(2(222)=444\).
time = 0.44, size = 15956, normalized size = 64.86
| | |
| method |
result |
size |
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| default |
\(\text {Expression too large to display}\) |
\(15956\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+a)^(5/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)
[Out]
result too large to display
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 5648 vs.
\(2 (226) = 452\).
time = 0.60, size = 5648, normalized size = 22.96 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="maxima")
[Out]
-5/32*c^9*d^10*arcsinh(c*x/sqrt(a*c))/(c^(13/2)*d^12*e^6 + 6*a*c^(11/2)*d^10*e^8 + 15*a^2*c^(9/2)*d^8*e^10 + 2
0*a^3*c^(7/2)*d^6*e^12 + 15*a^4*c^(5/2)*d^4*e^14 + 6*a^5*c^(3/2)*d^2*e^16 + a^6*sqrt(c)*e^18) - 5/32*a*c^8*d^8
*arcsinh(c*x/sqrt(a*c))/(c^(13/2)*d^12*e^4 + 6*a*c^(11/2)*d^10*e^6 + 15*a^2*c^(9/2)*d^8*e^8 + 20*a^3*c^(7/2)*d
^6*e^10 + 15*a^4*c^(5/2)*d^4*e^12 + 6*a^5*c^(3/2)*d^2*e^14 + a^6*sqrt(c)*e^16) + 5/32*sqrt(c*x^2 + a)*c^8*d^8*
x/(c^6*d^12*e^4 + 6*a*c^5*d^10*e^6 + 15*a^2*c^4*d^8*e^8 + 20*a^3*c^3*d^6*e^10 + 15*a^4*c^2*d^4*e^12 + 6*a^5*c*
d^2*e^14 + a^6*e^16) + 15/16*c^8*d^8*arcsinh(c*x/sqrt(a*c))/(c^(11/2)*d^10*e^6 + 5*a*c^(9/2)*d^8*e^8 + 10*a^2*
c^(7/2)*d^6*e^10 + 10*a^3*c^(5/2)*d^4*e^12 + 5*a^4*c^(3/2)*d^2*e^14 + a^5*sqrt(c)*e^16) - 5/48*(c*x^2 + a)^(3/
2)*c^7*d^7/(c^6*d^12*e^3 + 6*a*c^5*d^10*e^5 + 15*a^2*c^4*d^8*e^7 + 20*a^3*c^3*d^6*e^9 + 15*a^4*c^2*d^4*e^11 +
6*a^5*c*d^2*e^13 + a^6*e^15) + 5/48*(c*x^2 + a)^(3/2)*c^7*d^6*x/(c^6*d^12*e^2 + 6*a*c^5*d^10*e^4 + 15*a^2*c^4*
d^8*e^6 + 20*a^3*c^3*d^6*e^8 + 15*a^4*c^2*d^4*e^10 + 6*a^5*c*d^2*e^12 + a^6*e^14) + 5/32*sqrt(c*x^2 + a)*a*c^7
*d^6*x/(c^6*d^12*e^2 + 6*a*c^5*d^10*e^4 + 15*a^2*c^4*d^8*e^6 + 20*a^3*c^3*d^6*e^8 + 15*a^4*c^2*d^4*e^10 + 6*a^
5*c*d^2*e^12 + a^6*e^14) + 25/32*a*c^7*d^6*arcsinh(c*x/sqrt(a*c))/(c^(11/2)*d^10*e^4 + 5*a*c^(9/2)*d^8*e^6 + 1
0*a^2*c^(7/2)*d^6*e^8 + 10*a^3*c^(5/2)*d^4*e^10 + 5*a^4*c^(3/2)*d^2*e^12 + a^5*sqrt(c)*e^14) - 1/16*(c*x^2 + a
)^(5/2)*c^6*d^6/(c^6*d^12*x*e^2 + c^6*d^13*e + 6*a*c^5*d^10*x*e^4 + 6*a*c^5*d^11*e^3 + 15*a^2*c^4*d^8*x*e^6 +
15*a^2*c^4*d^9*e^5 + 20*a^3*c^3*d^6*x*e^8 + 20*a^3*c^3*d^7*e^7 + 15*a^4*c^2*d^4*x*e^10 + 15*a^4*c^2*d^5*e^9 +
6*a^5*c*d^2*x*e^12 + 6*a^5*c*d^3*e^11 + a^6*x*e^14 + a^6*d*e^13) - 5/16*sqrt(c*x^2 + a)*c^7*d^7/(c^5*d^10*e^5
+ 5*a*c^4*d^8*e^7 + 10*a^2*c^3*d^6*e^9 + 10*a^3*c^2*d^4*e^11 + 5*a^4*c*d^2*e^13 + a^5*e^15) - 15/32*sqrt(c*x^2
+ a)*c^7*d^6*x/(c^5*d^10*e^4 + 5*a*c^4*d^8*e^6 + 10*a^2*c^3*d^6*e^8 + 10*a^3*c^2*d^4*e^10 + 5*a^4*c*d^2*e^12
+ a^5*e^14) - 15/8*c^7*d^6*arcsinh(c*x/sqrt(a*c))/(c^(9/2)*d^8*e^6 + 4*a*c^(7/2)*d^6*e^8 + 6*a^2*c^(5/2)*d^4*e
^10 + 4*a^3*c^(3/2)*d^2*e^12 + a^4*sqrt(c)*e^14) + 1/48*(c*x^2 + a)^(7/2)*c^5*d^5/(c^6*d^12*x^2*e + c^6*d^14*e
^(-1) + 2*c^6*d^13*x + 6*a*c^5*d^10*x^2*e^3 + 12*a*c^5*d^11*x*e^2 + 6*a*c^5*d^12*e + 15*a^2*c^4*d^8*x^2*e^5 +
30*a^2*c^4*d^9*x*e^4 + 15*a^2*c^4*d^10*e^3 + 20*a^3*c^3*d^6*x^2*e^7 + 40*a^3*c^3*d^7*x*e^6 + 20*a^3*c^3*d^8*e^
5 + 15*a^4*c^2*d^4*x^2*e^9 + 30*a^4*c^2*d^5*x*e^8 + 15*a^4*c^2*d^6*e^7 + 6*a^5*c*d^2*x^2*e^11 + 12*a^5*c*d^3*x
*e^10 + 6*a^5*c*d^4*e^9 + a^6*x^2*e^13 + 2*a^6*d*x*e^12 + a^6*d^2*e^11) - 1/48*(c*x^2 + a)^(5/2)*c^6*d^5/(c^6*
d^12*e + 6*a*c^5*d^10*e^3 + 15*a^2*c^4*d^8*e^5 + 20*a^3*c^3*d^6*e^7 + 15*a^4*c^2*d^4*e^9 + 6*a^5*c*d^2*e^11 +
a^6*e^13) + 5/16*(c*x^2 + a)^(3/2)*c^6*d^5/(c^5*d^10*e^3 + 5*a*c^4*d^8*e^5 + 10*a^2*c^3*d^6*e^7 + 10*a^3*c^2*d
^4*e^9 + 5*a^4*c*d^2*e^11 + a^5*e^13) - 5/12*(c*x^2 + a)^(3/2)*c^6*d^4*x/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*
a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) - 5/8*sqrt(c*x^2 + a)*a*c^6*d^4*x/(c^5*d^1
0*e^2 + 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) - 5/16*c^7*d^
7*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-15)/(c*d^2*e^(-2) + a)^(9/2) - 35
/32*a*c^6*d^4*arcsinh(c*x/sqrt(a*c))/(c^(9/2)*d^8*e^4 + 4*a*c^(7/2)*d^6*e^6 + 6*a^2*c^(5/2)*d^4*e^8 + 4*a^3*c^
(3/2)*d^2*e^10 + a^4*sqrt(c)*e^12) - 1/24*(c*x^2 + a)^(7/2)*c^4*d^4/(c^5*d^10*x^3*e^2 + 3*c^5*d^11*x^2*e + c^5
*d^13*e^(-1) + 3*c^5*d^12*x + 5*a*c^4*d^8*x^3*e^4 + 15*a*c^4*d^9*x^2*e^3 + 15*a*c^4*d^10*x*e^2 + 5*a*c^4*d^11*
e + 10*a^2*c^3*d^6*x^3*e^6 + 30*a^2*c^3*d^7*x^2*e^5 + 30*a^2*c^3*d^8*x*e^4 + 10*a^2*c^3*d^9*e^3 + 10*a^3*c^2*d
^4*x^3*e^8 + 30*a^3*c^2*d^5*x^2*e^7 + 30*a^3*c^2*d^6*x*e^6 + 10*a^3*c^2*d^7*e^5 + 5*a^4*c*d^2*x^3*e^10 + 15*a^
4*c*d^3*x^2*e^9 + 15*a^4*c*d^4*x*e^8 + 5*a^4*c*d^5*e^7 + a^5*x^3*e^12 + 3*a^5*d*x^2*e^11 + 3*a^5*d^2*x*e^10 +
a^5*d^3*e^9) + 5/24*(c*x^2 + a)^(5/2)*c^5*d^4/(c^5*d^10*x*e^2 + c^5*d^11*e + 5*a*c^4*d^8*x*e^4 + 5*a*c^4*d^9*e
^3 + 10*a^2*c^3*d^6*x*e^6 + 10*a^2*c^3*d^7*e^5 + 10*a^3*c^2*d^4*x*e^8 + 10*a^3*c^2*d^5*e^7 + 5*a^4*c*d^2*x*e^1
0 + 5*a^4*c*d^3*e^9 + a^5*x*e^12 + a^5*d*e^11) + 15/16*sqrt(c*x^2 + a)*c^6*d^5/(c^4*d^8*e^5 + 4*a*c^3*d^6*e^7
+ 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13) + 15/32*sqrt(c*x^2 + a)*c^6*d^4*x/(c^4*d^8*e^4 + 4*a*c^3*d^
6*e^6 + 6*a^2*c^2*d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12) + 25/16*c^6*d^4*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e
^6 + 3*a*c^(5/2)*d^4*e^8 + 3*a^2*c^(3/2)*d^2*e^10 + a^3*sqrt(c)*e^12) - 1/8*(c*x^2 + a)^(7/2)*c^4*d^3/(c^5*d^1
0*x^2*e + c^5*d^12*e^(-1) + 2*c^5*d^11*x + 5*a*c^4*d^8*x^2*e^3 + 10*a*c^4*d^9*x*e^2 + 5*a*c^4*d^10*e + 10*a^2*
c^3*d^6*x^2*e^5 + 20*a^2*c^3*d^7*x*e^4 + 10*a^2*c^3*d^8*e^3 + 10*a^3*c^2*d^4*x^2*e^7 + 20*a^3*c^2*d^5*x*e^6 +
10*a^3*c^2*d^6*e^5 + 5*a^4*c*d^2*x^2*e^9 + 10*a^4*c*d^3*x*e^8 + 5*a^4*c*d^4*e^7 + a^5*x^2*e^11 + 2*a^5*d*x*e^1
0 + a^5*d^2*e^9) + 1/8*(c*x^2 + a)^(5/2)*c^5*d^...
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1283 vs.
\(2 (226) = 452\).
time = 42.51, size = 2593, normalized size = 10.54 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="fricas")
[Out]
[1/672*(105*(a^3*c^4*d*x^7*e^7 + 7*a^3*c^4*d^2*x^6*e^6 + 21*a^3*c^4*d^3*x^5*e^5 + 35*a^3*c^4*d^4*x^4*e^4 + 35*
a^3*c^4*d^5*x^3*e^3 + 21*a^3*c^4*d^6*x^2*e^2 + 7*a^3*c^4*d^7*x*e + a^3*c^4*d^8)*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*(56*c^7*d^9*x^5 + 182*a*c^6*d^9*x^3 + 231*a^2*c^5*d^9*x - 48*(a^4*c^3*x^6 +
3*a^5*c^2*x^4 + 3*a^6*c*x^2 + a^7)*e^9 - 7*(33*a^4*c^3*d*x^5 + 26*a^5*c^2*d*x^3 + 8*a^6*c*d*x)*e^8 + (39*a^3*c
^4*d^2*x^6 - 1185*a^4*c^3*d^2*x^4 - 842*a^5*c^2*d^2*x^2 - 248*a^6*c*d^2)*e^7 + 14*(27*a^3*c^4*d^3*x^5 - 101*a^
4*c^3*d^3*x^3 - 23*a^5*c^2*d^3*x)*e^6 + (125*a^2*c^5*d^4*x^6 - 241*a^3*c^4*d^4*x^4 - 2362*a^4*c^3*d^4*x^2 - 52
6*a^5*c^2*d^4)*e^5 + 875*(a^2*c^5*d^5*x^5 - a^4*c^3*d^5*x)*e^4 + (46*a*c^6*d^6*x^6 + 922*a^2*c^5*d^6*x^4 - 119
9*a^3*c^4*d^6*x^2 - 605*a^4*c^3*d^6)*e^3 + 14*(23*a*c^6*d^7*x^5 + 101*a^2*c^5*d^7*x^3 - 27*a^3*c^4*d^7*x)*e^2
+ (8*c^7*d^8*x^6 + 122*a*c^6*d^8*x^4 + 465*a^2*c^5*d^8*x^2 - 279*a^3*c^4*d^8)*e)*sqrt(c*x^2 + a))/(7*c^5*d^16*
x*e + c^5*d^17 + a^5*x^7*e^17 + 7*a^5*d*x^6*e^16 + (5*a^4*c*d^2*x^7 + 21*a^5*d^2*x^5)*e^15 + 35*(a^4*c*d^3*x^6
+ a^5*d^3*x^4)*e^14 + 5*(2*a^3*c^2*d^4*x^7 + 21*a^4*c*d^4*x^5 + 7*a^5*d^4*x^3)*e^13 + 7*(10*a^3*c^2*d^5*x^6 +
25*a^4*c*d^5*x^4 + 3*a^5*d^5*x^2)*e^12 + (10*a^2*c^3*d^6*x^7 + 210*a^3*c^2*d^6*x^5 + 175*a^4*c*d^6*x^3 + 7*a^
5*d^6*x)*e^11 + (70*a^2*c^3*d^7*x^6 + 350*a^3*c^2*d^7*x^4 + 105*a^4*c*d^7*x^2 + a^5*d^7)*e^10 + 5*(a*c^4*d^8*x
^7 + 42*a^2*c^3*d^8*x^5 + 70*a^3*c^2*d^8*x^3 + 7*a^4*c*d^8*x)*e^9 + 5*(7*a*c^4*d^9*x^6 + 70*a^2*c^3*d^9*x^4 +
42*a^3*c^2*d^9*x^2 + a^4*c*d^9)*e^8 + (c^5*d^10*x^7 + 105*a*c^4*d^10*x^5 + 350*a^2*c^3*d^10*x^3 + 70*a^3*c^2*d
^10*x)*e^7 + (7*c^5*d^11*x^6 + 175*a*c^4*d^11*x^4 + 210*a^2*c^3*d^11*x^2 + 10*a^3*c^2*d^11)*e^6 + 7*(3*c^5*d^1
2*x^5 + 25*a*c^4*d^12*x^3 + 10*a^2*c^3*d^12*x)*e^5 + 5*(7*c^5*d^13*x^4 + 21*a*c^4*d^13*x^2 + 2*a^2*c^3*d^13)*e
^4 + 35*(c^5*d^14*x^3 + a*c^4*d^14*x)*e^3 + (21*c^5*d^15*x^2 + 5*a*c^4*d^15)*e^2), 1/336*(105*(a^3*c^4*d*x^7*e
^7 + 7*a^3*c^4*d^2*x^6*e^6 + 21*a^3*c^4*d^3*x^5*e^5 + 35*a^3*c^4*d^4*x^4*e^4 + 35*a^3*c^4*d^5*x^3*e^3 + 21*a^3
*c^4*d^6*x^2*e^2 + 7*a^3*c^4*d^7*x*e + a^3*c^4*d^8)*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)) + (56*c^7*d^9*x^5 + 182*a*c^6*d^9*x^3 + 2
31*a^2*c^5*d^9*x - 48*(a^4*c^3*x^6 + 3*a^5*c^2*x^4 + 3*a^6*c*x^2 + a^7)*e^9 - 7*(33*a^4*c^3*d*x^5 + 26*a^5*c^2
*d*x^3 + 8*a^6*c*d*x)*e^8 + (39*a^3*c^4*d^2*x^6 - 1185*a^4*c^3*d^2*x^4 - 842*a^5*c^2*d^2*x^2 - 248*a^6*c*d^2)*
e^7 + 14*(27*a^3*c^4*d^3*x^5 - 101*a^4*c^3*d^3*x^3 - 23*a^5*c^2*d^3*x)*e^6 + (125*a^2*c^5*d^4*x^6 - 241*a^3*c^
4*d^4*x^4 - 2362*a^4*c^3*d^4*x^2 - 526*a^5*c^2*d^4)*e^5 + 875*(a^2*c^5*d^5*x^5 - a^4*c^3*d^5*x)*e^4 + (46*a*c^
6*d^6*x^6 + 922*a^2*c^5*d^6*x^4 - 1199*a^3*c^4*d^6*x^2 - 605*a^4*c^3*d^6)*e^3 + 14*(23*a*c^6*d^7*x^5 + 101*a^2
*c^5*d^7*x^3 - 27*a^3*c^4*d^7*x)*e^2 + (8*c^7*d^8*x^6 + 122*a*c^6*d^8*x^4 + 465*a^2*c^5*d^8*x^2 - 279*a^3*c^4*
d^8)*e)*sqrt(c*x^2 + a))/(7*c^5*d^16*x*e + c^5*d^17 + a^5*x^7*e^17 + 7*a^5*d*x^6*e^16 + (5*a^4*c*d^2*x^7 + 21*
a^5*d^2*x^5)*e^15 + 35*(a^4*c*d^3*x^6 + a^5*d^3*x^4)*e^14 + 5*(2*a^3*c^2*d^4*x^7 + 21*a^4*c*d^4*x^5 + 7*a^5*d^
4*x^3)*e^13 + 7*(10*a^3*c^2*d^5*x^6 + 25*a^4*c*d^5*x^4 + 3*a^5*d^5*x^2)*e^12 + (10*a^2*c^3*d^6*x^7 + 210*a^3*c
^2*d^6*x^5 + 175*a^4*c*d^6*x^3 + 7*a^5*d^6*x)*e^11 + (70*a^2*c^3*d^7*x^6 + 350*a^3*c^2*d^7*x^4 + 105*a^4*c*d^7
*x^2 + a^5*d^7)*e^10 + 5*(a*c^4*d^8*x^7 + 42*a^2*c^3*d^8*x^5 + 70*a^3*c^2*d^8*x^3 + 7*a^4*c*d^8*x)*e^9 + 5*(7*
a*c^4*d^9*x^6 + 70*a^2*c^3*d^9*x^4 + 42*a^3*c^2*d^9*x^2 + a^4*c*d^9)*e^8 + (c^5*d^10*x^7 + 105*a*c^4*d^10*x^5
+ 350*a^2*c^3*d^10*x^3 + 70*a^3*c^2*d^10*x)*e^7 + (7*c^5*d^11*x^6 + 175*a*c^4*d^11*x^4 + 210*a^2*c^3*d^11*x^2
+ 10*a^3*c^2*d^11)*e^6 + 7*(3*c^5*d^12*x^5 + 25*a*c^4*d^12*x^3 + 10*a^2*c^3*d^12*x)*e^5 + 5*(7*c^5*d^13*x^4 +
21*a*c^4*d^13*x^2 + 2*a^2*c^3*d^13)*e^4 + 35*(c^5*d^14*x^3 + a*c^4*d^14*x)*e^3 + (21*c^5*d^15*x^2 + 5*a*c^4*d^
15)*e^2)]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+a)**(5/2)/(e*x+d)**8,x)
[Out]
Integral((a + c*x**2)**(5/2)/(d + e*x)**8, x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2354 vs.
\(2 (226) = 452\).
time = 2.01, size = 2354, normalized size = 9.57 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="giac")
[Out]
-5/8*a^3*c^4*d*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^4*d^8 + 4*a*c^3*
d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d^2 - a*e^2)) + 1/168*(1792*(sqrt(c)*x - sqrt
(c*x^2 + a))^8*c^(19/2)*d^12*e + 512*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^10*d^13 + 2688*(sqrt(c)*x - sqrt(c*x^2
+ a))^9*c^9*d^11*e^2 + 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*d^10*e^3 - 1792*(sqrt(c)*x - sqrt(c*x^2
+ a))^6*a*c^(19/2)*d^12*e + 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^11*c^8*d^9*e^4 - 2944*(sqrt(c)*x - sqrt(c*x^2 +
a))^7*a*c^9*d^11*e^2 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*c^(15/2)*d^8*e^5 + 1792*(sqrt(c)*x - sqrt(c*x^2 +
a))^8*a*c^(17/2)*d^10*e^3 + 8288*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^8*d^9*e^4 + 2688*(sqrt(c)*x - sqrt(c*x^2
+ a))^5*a^2*c^9*d^11*e^2 + 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/2)*d^8*e^5 - 1792*(sqrt(c)*x - sqrt(
c*x^2 + a))^6*a^2*c^(17/2)*d^10*e^3 + 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*d^7*e^6 - 13248*(sqrt(c)*x -
sqrt(c*x^2 + a))^7*a^2*c^8*d^9*e^4 + 1344*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a*c^(13/2)*d^6*e^7 - 9072*(sqrt(c)
*x - sqrt(c*x^2 + a))^8*a^2*c^(15/2)*d^8*e^5 - 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(17/2)*d^10*e^3 + 62
72*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*d^7*e^6 + 8288*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^8*d^9*e^4 + 13
440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*d^6*e^7 + 9072*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(15/2)*
d^8*e^5 + 6720*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*d^5*e^8 - 30736*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^
7*d^7*e^6 + 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^8*d^9*e^4 + 2016*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^2*c
^(11/2)*d^4*e^9 - 55832*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(13/2)*d^6*e^7 - 8960*(sqrt(c)*x - sqrt(c*x^2 +
a))^4*a^4*c^(15/2)*d^8*e^5 - 42588*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^6*d^5*e^8 + 11312*(sqrt(c)*x - sqrt(c
*x^2 + a))^5*a^4*c^7*d^7*e^6 - 13370*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(11/2)*d^4*e^9 + 80192*(sqrt(c)*x
- sqrt(c*x^2 + a))^6*a^4*c^(13/2)*d^6*e^7 - 336*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(15/2)*d^8*e^5 - 3010*(s
qrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*d^3*e^10 + 100016*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^6*d^5*e^8 + 548
8*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^7*d^7*e^6 - 21*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)*d^2*e^11 +
70210*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(11/2)*d^4*e^9 - 19488*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(13/
2)*d^6*e^7 - 105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*d*e^12 + 27370*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c
^5*d^3*e^10 - 79128*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^6*d^5*e^8 + 112*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^
7*d^7*e^6 + 9940*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(9/2)*d^2*e^11 - 82180*(sqrt(c)*x - sqrt(c*x^2 + a))^6
*a^5*c^(11/2)*d^4*e^9 - 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(13/2)*d^6*e^7 + 1820*(sqrt(c)*x - sqrt(c*x
^2 + a))^11*a^4*c^4*d*e^12 - 52500*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^5*d^3*e^10 + 14448*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*a^6*c^6*d^5*e^8 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*e^13 - 16485*(sqrt(c)*x - sqr
t(c*x^2 + a))^8*a^5*c^(9/2)*d^2*e^11 + 49252*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(11/2)*d^4*e^9 - 8*a^7*c^(1
3/2)*d^6*e^7 - 4445*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^5*c^4*d*e^12 + 44660*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6
*c^5*d^3*e^10 + 532*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^6*d^5*e^8 + 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*
c^(9/2)*d^2*e^11 - 5026*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(11/2)*d^4*e^9 + 6720*(sqrt(c)*x - sqrt(c*x^2 +
a))^7*a^6*c^4*d*e^12 - 17738*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^5*d^3*e^10 + 1680*(sqrt(c)*x - sqrt(c*x^2 +
a))^8*a^6*c^(7/2)*e^13 - 12047*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(9/2)*d^2*e^11 - 38*a^8*c^(11/2)*d^4*e^9
- 5635*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^4*d*e^12 + 1218*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^5*d^3*e^10 +
4620*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(9/2)*d^2*e^11 + 2212*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^8*c^4*d*e^
12 + 1008*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^8*c^(7/2)*e^13 - 87*a^9*c^(9/2)*d^2*e^11 - 567*(sqrt(c)*x - sqrt(c
*x^2 + a))*a^9*c^4*d*e^12 + 48*a^10*c^(7/2)*e^13)/((c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 + 4*a^3
*c*d^2*e^12 + a^4*e^14)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^
7)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + c*x^2)^(5/2)/(d + e*x)^8,x)
[Out]
int((a + c*x^2)^(5/2)/(d + e*x)^8, x)
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