3.6.44 \(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^7} \, dx\) [544]
Optimal. Leaf size=269 \[ -\frac {a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {a^2 c^3 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]
[Out]
-1/24*c*(-a*e^2+6*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^3/(e*x+d)^4-1/6*e*(c*x^2+a)^(5/2)/(a*e^2+c
*d^2)/(e*x+d)^6-7/30*c*d*e*(c*x^2+a)^(5/2)/(a*e^2+c*d^2)^2/(e*x+d)^5-1/16*a^2*c^3*(-a*e^2+6*c*d^2)*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)-1/16*a*c^2*(-a*e^2+6*c*d^2)*(-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 = 0.14, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {759, 821, 735,
739, 212} \begin {gather*} -\frac {a^2 c^3 \left (6 c d^2-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 )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac {a c^2 \sqrt {a+c x^2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 (d+e x)^5 \left (a e^2+c d^2\right )^2}-\frac {c \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + c*x^2)^(3/2)/(d + e*x)^7,x]
[Out]
-1/16*(a*c^2*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)^4*(d + e*x)^2) - (c*(6*c*d^2 -
a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(6*(c*d^2 +
a*e^2)*(d + e*x)^6) - (7*c*d*e*(a + c*x^2)^(5/2))/(30*(c*d^2 + a*e^2)^2*(d + e*x)^5) - (a^2*c^3*(6*c*d^2 - a*
e^2)*ArcTanh[(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))
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 759
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 821
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]
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx &=-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {c \int \frac {(-6 d+e x) \left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{6 \left (c d^2+a e^2\right )}\\ &=-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac {\left (c \left (6 c d^2-a e^2\right )\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 {c \left (6 c d^2-a e^2\right ) (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 {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac {\left (a c^2 \left (6 c d^2-a e^2\right )\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac {a c^2 \left (6 c d^2-a e^2\right ) (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 {c \left (6 c d^2-a e^2\right ) (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 {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac {\left (a^2 c^3 \left (6 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac {a c^2 \left (6 c d^2-a e^2\right ) (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 {c \left (6 c d^2-a e^2\right ) (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 {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {\left (a^2 c^3 \left (6 c d^2-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 )}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac {a c^2 \left (6 c d^2-a e^2\right ) (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 {c \left (6 c d^2-a e^2\right ) (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 {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {a^2 c^3 \left (6 c d^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 )}{16 \left (c d^2+a e^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 10.51, size = 358, normalized size = 1.33 \begin {gather*} \frac {1}{240} \left (-\frac {\sqrt {a+c x^2} \left (40 \left (c d^2+a e^2\right )^5-104 c d \left (c d^2+a e^2\right )^4 (d+e x)+2 c \left (c d^2+a e^2\right )^3 \left (38 c d^2+35 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right )^2 \left (2 c d^2+9 a e^2\right ) (d+e x)^3-c^2 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) (d+e x)^4-c^3 d \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) (d+e x)^5\right )}{e^3 \left (c d^2+a e^2\right )^4 (d+e x)^6}+\frac {15 a^2 c^3 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{9/2}}+\frac {15 a^2 c^3 \left (-6 c d^2+a e^2\right ) \log \left (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 )^{9/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + c*x^2)^(3/2)/(d + e*x)^7,x]
[Out]
(-((Sqrt[a + c*x^2]*(40*(c*d^2 + a*e^2)^5 - 104*c*d*(c*d^2 + a*e^2)^4*(d + e*x) + 2*c*(c*d^2 + a*e^2)^3*(38*c*
d^2 + 35*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^2*(2*c*d^2 + 9*a*e^2)*(d + e*x)^3 - c^2*(c*d^2 + a*e^2)*
(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*(d + e*x)^4 - c^3*d*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*(d + e
*x)^5))/(e^3*(c*d^2 + a*e^2)^4*(d + e*x)^6)) + (15*a^2*c^3*(6*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/
2) + (15*a^2*c^3*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9
/2))/240
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8230\) vs.
\(2(245)=490\).
time = 0.63, size = 8231, normalized size = 30.60
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(8231\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+a)^(3/2)/(e*x+d)^7,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. 3132 vs.
\(2 (246) = 492\).
time = 0.48, size = 3132, normalized size = 11.64 \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)^(3/2)/(e*x+d)^7,x, algorithm="maxima")
[Out]
7/16*sqrt(c*x^2 + a)*c^6*d^6/(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) - 7/16*sqrt(c*x^2 + a)*c^6*d^5*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) + 7/48*(c*x^2 + a)^(3/2)*c^5*d^5/(c^5*d^10*x*e^2 + c^5*d^1
1*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^10 + 5*a^4*c*d^3*e^9 + a^5*x*e^12 + a^5*d*e^11) - 7/48*(c*x^2 + a)^(5/2)
*c^4*d^4/(c^5*d^10*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^1
1 + 2*a^5*d*x*e^10 + a^5*d^2*e^9) + 7/48*(c*x^2 + a)^(3/2)*c^5*d^4/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3*
d^6*e^5 + 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 + a^5*e^11) + 7/16*c^6*d^6*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d
)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-13)/(c*d^2*e^(-2) + a)^(9/2) - 15/16*sqrt(c*x^2 + a)*c^5*d^4/(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) + 1/2*sqrt(c*x^2 + a)*c^5*d^3*x/(c^4*d^
8*e^2 + 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10) - 7/24*(c*x^2 + a)^(5/2)*c^3*d^3/(c^
4*d^8*x^3*e^2 + 3*c^4*d^9*x^2*e + c^4*d^11*e^(-1) + 3*c^4*d^10*x + 4*a*c^3*d^6*x^3*e^4 + 12*a*c^3*d^7*x^2*e^3
+ 12*a*c^3*d^8*x*e^2 + 4*a*c^3*d^9*e + 6*a^2*c^2*d^4*x^3*e^6 + 18*a^2*c^2*d^5*x^2*e^5 + 18*a^2*c^2*d^6*x*e^4 +
6*a^2*c^2*d^7*e^3 + 4*a^3*c*d^2*x^3*e^8 + 12*a^3*c*d^3*x^2*e^7 + 12*a^3*c*d^4*x*e^6 + 4*a^3*c*d^5*e^5 + a^4*x
^3*e^10 + 3*a^4*d*x^2*e^9 + 3*a^4*d^2*x*e^8 + a^4*d^3*e^7) - 11/24*(c*x^2 + a)^(3/2)*c^4*d^3/(c^4*d^8*x*e^2 +
c^4*d^9*e + 4*a*c^3*d^6*x*e^4 + 4*a*c^3*d^7*e^3 + 6*a^2*c^2*d^4*x*e^6 + 6*a^2*c^2*d^5*e^5 + 4*a^3*c*d^2*x*e^8
+ 4*a^3*c*d^3*e^7 + a^4*x*e^10 + a^4*d*e^9) - 15/16*c^5*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt
(a*c)*abs(x*e + d)))*e^(-11)/(c*d^2*e^(-2) + a)^(7/2) - 1/8*(c*x^2 + a)^(5/2)*c^3*d^2/(c^4*d^8*x^2*e + c^4*d^1
0*e^(-1) + 2*c^4*d^9*x + 4*a*c^3*d^6*x^2*e^3 + 8*a*c^3*d^7*x*e^2 + 4*a*c^3*d^8*e + 6*a^2*c^2*d^4*x^2*e^5 + 12*
a^2*c^2*d^5*x*e^4 + 6*a^2*c^2*d^6*e^3 + 4*a^3*c*d^2*x^2*e^7 + 8*a^3*c*d^3*x*e^6 + 4*a^3*c*d^4*e^5 + a^4*x^2*e^
9 + 2*a^4*d*x*e^8 + a^4*d^2*e^7) + 1/8*(c*x^2 + a)^(3/2)*c^4*d^2/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*
e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) - 7/24*(c*x^2 + a)^(5/2)*c^2*d^2/(c^3*d^6*x^4*e^3 + 4*c^3*d^7*x^3*e^2 + 6*c^3
*d^8*x^2*e + c^3*d^10*e^(-1) + 4*c^3*d^9*x + 3*a*c^2*d^4*x^4*e^5 + 12*a*c^2*d^5*x^3*e^4 + 18*a*c^2*d^6*x^2*e^3
+ 12*a*c^2*d^7*x*e^2 + 3*a*c^2*d^8*e + 3*a^2*c*d^2*x^4*e^7 + 12*a^2*c*d^3*x^3*e^6 + 18*a^2*c*d^4*x^2*e^5 + 12
*a^2*c*d^5*x*e^4 + 3*a^2*c*d^6*e^3 + a^3*x^4*e^9 + 4*a^3*d*x^3*e^8 + 6*a^3*d^2*x^2*e^7 + 4*a^3*d^3*x*e^6 + a^3
*d^4*e^5) + 9/16*sqrt(c*x^2 + a)*c^4*d^2/(c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*e^9) - 1/16*sq
rt(c*x^2 + a)*c^4*d*x/(c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 + a^3*e^8) + 9/16*c^4*d^2*arcsinh(c*d*x
/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-9)/(c*d^2*e^(-2) + a)^(5/2) + 1/24*(c*x^2 + a)^(
5/2)*c^2*d/(c^3*d^6*x^3*e^2 + 3*c^3*d^7*x^2*e + c^3*d^9*e^(-1) + 3*c^3*d^8*x + 3*a*c^2*d^4*x^3*e^4 + 9*a*c^2*d
^5*x^2*e^3 + 9*a*c^2*d^6*x*e^2 + 3*a*c^2*d^7*e + 3*a^2*c*d^2*x^3*e^6 + 9*a^2*c*d^3*x^2*e^5 + 9*a^2*c*d^4*x*e^4
+ 3*a^2*c*d^5*e^3 + a^3*x^3*e^8 + 3*a^3*d*x^2*e^7 + 3*a^3*d^2*x*e^6 + a^3*d^3*e^5) + 1/16*(c*x^2 + a)^(3/2)*c
^3*d/(c^3*d^6*x*e^2 + c^3*d^7*e + 3*a*c^2*d^4*x*e^4 + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^2*x*e^6 + 3*a^2*c*d^3*e^5 +
a^3*x*e^8 + a^3*d*e^7) + 1/48*(c*x^2 + a)^(5/2)*c^2/(c^3*d^6*x^2*e + c^3*d^8*e^(-1) + 2*c^3*d^7*x + 3*a*c^2*d^
4*x^2*e^3 + 6*a*c^2*d^5*x*e^2 + 3*a*c^2*d^6*e + 3*a^2*c*d^2*x^2*e^5 + 6*a^2*c*d^3*x*e^4 + 3*a^2*c*d^4*e^3 + a^
3*x^2*e^7 + 2*a^3*d*x*e^6 + a^3*d^2*e^5) - 1/48*(c*x^2 + a)^(3/2)*c^3/(c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d
^2*e^5 + a^3*e^7) - 7/30*(c*x^2 + a)^(5/2)*c*d/(c^2*d^4*x^5*e^4 + 5*c^2*d^5*x^4*e^3 + 10*c^2*d^6*x^3*e^2 + 10*
c^2*d^7*x^2*e + c^2*d^9*e^(-1) + 5*c^2*d^8*x + 2*a*c*d^2*x^5*e^6 + 10*a*c*d^3*x^4*e^5 + 20*a*c*d^4*x^3*e^4 + 2
0*a*c*d^5*x^2*e^3 + 10*a*c*d^6*x*e^2 + 2*a*c*d^7*e + a^2*x^5*e^8 + 5*a^2*d*x^4*e^7 + 10*a^2*d^2*x^3*e^6 + 10*a
^2*d^3*x^2*e^5 + 5*a^2*d^4*x*e^4 + a^2*d^5*e^3) - 1/16*c^3*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(
a*c)*abs(x*e + d)))*e^(-7)/(c*d^2*e^(-2) + a)^(3/2) + 1/24*(c*x^2 + a)^(5/2)*c/(c^2*d^4*x^4*e^3 + 4*c^2*d^5*x^
3*e^2 + 6*c^2*d^6*x^2*e + c^2*d^8*e^(-1) + 4*c^2*d^7*x + 2*a*c*d^2*x^4*e^5 + 8*a*c*d^3*x^3*e^4 + 12*a*c*d^4*x^
2*e^3 + 8*a*c*d^5*x*e^2 + 2*a*c*d^6*e + a^2*x^4*e^7 + 4*a^2*d*x^3*e^6 + 6*a^2*d^2*x^2*e^5 + 4*a^2*d^3*x*e^4 +
a^2*d^4*e^3) - 1/16*sqrt(c*x^2 + a)*c^3/(c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7) - 1/6*(c*x^2 + a)^(5/2)/(c*d^2
*x^6*e^5 + 6*c*d^3*x^5*e^4 + 15*c*d^4*x^4*e^3 +...
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1238 vs.
\(2 (246) = 492\).
time = 22.69, size = 2503, normalized size = 9.30 \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)^(3/2)/(e*x+d)^7,x, algorithm="fricas")
[Out]
[-1/480*(15*(36*a^2*c^4*d^7*x*e + 6*a^2*c^4*d^8 - a^3*c^3*x^6*e^8 - 6*a^3*c^3*d*x^5*e^7 + 3*(2*a^2*c^4*d^2*x^6
- 5*a^3*c^3*d^2*x^4)*e^6 + 4*(9*a^2*c^4*d^3*x^5 - 5*a^3*c^3*d^3*x^3)*e^5 + 15*(6*a^2*c^4*d^4*x^4 - a^3*c^3*d^
4*x^2)*e^4 + 6*(20*a^2*c^4*d^5*x^3 - a^3*c^3*d^5*x)*e^3 + (90*a^2*c^4*d^6*x^2 - a^3*c^3*d^6)*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*(60*c^6*d^9*x^3 + 150*a*c^5*d^9*x - 5*(3*a^4*c^2*x^4 + 14*
a^5*c*x^2 + 8*a^6)*e^9 - 3*(27*a^3*c^3*d*x^5 + 14*a^4*c^2*d*x^3 + 12*a^5*c*d*x)*e^8 - (411*a^3*c^3*d^2*x^4 + 3
92*a^4*c^2*d^2*x^2 + 206*a^5*c*d^2)*e^7 - (53*a^2*c^4*d^3*x^5 + 776*a^3*c^3*d^3*x^3 + 198*a^4*c^2*d^3*x)*e^6 -
(228*a^2*c^4*d^4*x^4 + 1396*a^3*c^3*d^4*x^2 + 433*a^4*c^2*d^4)*e^5 + (32*a*c^5*d^5*x^5 - 316*a^2*c^4*d^5*x^3
- 663*a^3*c^3*d^5*x)*e^4 + 3*(64*a*c^5*d^6*x^4 - 282*a^2*c^4*d^6*x^2 - 171*a^3*c^3*d^6)*e^3 + (4*c^6*d^7*x^5 +
478*a*c^5*d^7*x^3 - 351*a^2*c^4*d^7*x)*e^2 + 6*(4*c^6*d^8*x^4 + 38*a*c^5*d^8*x^2 - 41*a^2*c^4*d^8)*e)*sqrt(c*
x^2 + a))/(6*c^5*d^15*x*e + c^5*d^16 + a^5*x^6*e^16 + 6*a^5*d*x^5*e^15 + 5*(a^4*c*d^2*x^6 + 3*a^5*d^2*x^4)*e^1
4 + 10*(3*a^4*c*d^3*x^5 + 2*a^5*d^3*x^3)*e^13 + 5*(2*a^3*c^2*d^4*x^6 + 15*a^4*c*d^4*x^4 + 3*a^5*d^4*x^2)*e^12
+ 2*(30*a^3*c^2*d^5*x^5 + 50*a^4*c*d^5*x^3 + 3*a^5*d^5*x)*e^11 + (10*a^2*c^3*d^6*x^6 + 150*a^3*c^2*d^6*x^4 + 7
5*a^4*c*d^6*x^2 + a^5*d^6)*e^10 + 10*(6*a^2*c^3*d^7*x^5 + 20*a^3*c^2*d^7*x^3 + 3*a^4*c*d^7*x)*e^9 + 5*(a*c^4*d
^8*x^6 + 30*a^2*c^3*d^8*x^4 + 30*a^3*c^2*d^8*x^2 + a^4*c*d^8)*e^8 + 10*(3*a*c^4*d^9*x^5 + 20*a^2*c^3*d^9*x^3 +
6*a^3*c^2*d^9*x)*e^7 + (c^5*d^10*x^6 + 75*a*c^4*d^10*x^4 + 150*a^2*c^3*d^10*x^2 + 10*a^3*c^2*d^10)*e^6 + 2*(3
*c^5*d^11*x^5 + 50*a*c^4*d^11*x^3 + 30*a^2*c^3*d^11*x)*e^5 + 5*(3*c^5*d^12*x^4 + 15*a*c^4*d^12*x^2 + 2*a^2*c^3
*d^12)*e^4 + 10*(2*c^5*d^13*x^3 + 3*a*c^4*d^13*x)*e^3 + 5*(3*c^5*d^14*x^2 + a*c^4*d^14)*e^2), 1/240*(15*(36*a^
2*c^4*d^7*x*e + 6*a^2*c^4*d^8 - a^3*c^3*x^6*e^8 - 6*a^3*c^3*d*x^5*e^7 + 3*(2*a^2*c^4*d^2*x^6 - 5*a^3*c^3*d^2*x
^4)*e^6 + 4*(9*a^2*c^4*d^3*x^5 - 5*a^3*c^3*d^3*x^3)*e^5 + 15*(6*a^2*c^4*d^4*x^4 - a^3*c^3*d^4*x^2)*e^4 + 6*(20
*a^2*c^4*d^5*x^3 - a^3*c^3*d^5*x)*e^3 + (90*a^2*c^4*d^6*x^2 - a^3*c^3*d^6)*e^2)*sqrt(-c*d^2 - a*e^2)*arctan(-s
qrt(-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)) + (60*c^6*d^9
*x^3 + 150*a*c^5*d^9*x - 5*(3*a^4*c^2*x^4 + 14*a^5*c*x^2 + 8*a^6)*e^9 - 3*(27*a^3*c^3*d*x^5 + 14*a^4*c^2*d*x^3
+ 12*a^5*c*d*x)*e^8 - (411*a^3*c^3*d^2*x^4 + 392*a^4*c^2*d^2*x^2 + 206*a^5*c*d^2)*e^7 - (53*a^2*c^4*d^3*x^5 +
776*a^3*c^3*d^3*x^3 + 198*a^4*c^2*d^3*x)*e^6 - (228*a^2*c^4*d^4*x^4 + 1396*a^3*c^3*d^4*x^2 + 433*a^4*c^2*d^4)
*e^5 + (32*a*c^5*d^5*x^5 - 316*a^2*c^4*d^5*x^3 - 663*a^3*c^3*d^5*x)*e^4 + 3*(64*a*c^5*d^6*x^4 - 282*a^2*c^4*d^
6*x^2 - 171*a^3*c^3*d^6)*e^3 + (4*c^6*d^7*x^5 + 478*a*c^5*d^7*x^3 - 351*a^2*c^4*d^7*x)*e^2 + 6*(4*c^6*d^8*x^4
+ 38*a*c^5*d^8*x^2 - 41*a^2*c^4*d^8)*e)*sqrt(c*x^2 + a))/(6*c^5*d^15*x*e + c^5*d^16 + a^5*x^6*e^16 + 6*a^5*d*x
^5*e^15 + 5*(a^4*c*d^2*x^6 + 3*a^5*d^2*x^4)*e^14 + 10*(3*a^4*c*d^3*x^5 + 2*a^5*d^3*x^3)*e^13 + 5*(2*a^3*c^2*d^
4*x^6 + 15*a^4*c*d^4*x^4 + 3*a^5*d^4*x^2)*e^12 + 2*(30*a^3*c^2*d^5*x^5 + 50*a^4*c*d^5*x^3 + 3*a^5*d^5*x)*e^11
+ (10*a^2*c^3*d^6*x^6 + 150*a^3*c^2*d^6*x^4 + 75*a^4*c*d^6*x^2 + a^5*d^6)*e^10 + 10*(6*a^2*c^3*d^7*x^5 + 20*a^
3*c^2*d^7*x^3 + 3*a^4*c*d^7*x)*e^9 + 5*(a*c^4*d^8*x^6 + 30*a^2*c^3*d^8*x^4 + 30*a^3*c^2*d^8*x^2 + a^4*c*d^8)*e
^8 + 10*(3*a*c^4*d^9*x^5 + 20*a^2*c^3*d^9*x^3 + 6*a^3*c^2*d^9*x)*e^7 + (c^5*d^10*x^6 + 75*a*c^4*d^10*x^4 + 150
*a^2*c^3*d^10*x^2 + 10*a^3*c^2*d^10)*e^6 + 2*(3*c^5*d^11*x^5 + 50*a*c^4*d^11*x^3 + 30*a^2*c^3*d^11*x)*e^5 + 5*
(3*c^5*d^12*x^4 + 15*a*c^4*d^12*x^2 + 2*a^2*c^3*d^12)*e^4 + 10*(2*c^5*d^13*x^3 + 3*a*c^4*d^13*x)*e^3 + 5*(3*c^
5*d^14*x^2 + a*c^4*d^14)*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 {3}{2}}}{\left (d + e x\right )^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+a)**(3/2)/(e*x+d)**7,x)
[Out]
Integral((a + c*x**2)**(3/2)/(d + e*x)**7, x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1818 vs.
\(2 (246) = 492\).
time = 0.67, size = 1818, normalized size = 6.76 \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)^(3/2)/(e*x+d)^7,x, algorithm="giac")
[Out]
-1/8*(6*a^2*c^4*d^2 - a^3*c^3*e^2)*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)) + 1/120*(38
4*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*e + 128*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 + 480*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 + 320*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 384*(sqrt(c)*x -
sqrt(c*x^2 + a))^5*a*c^8*d^10*e - 64*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 + 1728*(sqrt(c)*x - s
qrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 1920*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 480*(sqrt(c)*x - s
qrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2 + 1280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 - 1728*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^3 - 8592*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 - 9456*(sqrt(
c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 320*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d^8*e^3 - 7380*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 + 3840*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*d^7*e^4 - 25
20*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*e^7 + 19056*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^5 - 9
90*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*d^3*e^8 + 24440*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d
^5*e^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(13/2)*d^7*e^4 - 90*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^
4*d^2*e^9 + 20760*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^5*d^4*e^7 - 2960*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c
^6*d^6*e^5 + 8220*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*d^3*e^8 - 18720*(sqrt(c)*x - sqrt(c*x^2 + a))^4*
a^4*c^(11/2)*d^5*e^6 + 2530*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*d^2*e^9 - 21480*(sqrt(c)*x - sqrt(c*x^2 +
a))^5*a^4*c^5*d^4*e^7 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^6*d^6*e^5 + 165*(sqrt(c)*x - sqrt(c*x^2 + a))^1
0*a^3*c^(7/2)*d*e^10 - 14860*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(9/2)*d^3*e^8 + 1656*(sqrt(c)*x - sqrt(c*x^
2 + a))^2*a^5*c^(11/2)*d^5*e^6 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*e^11 - 2700*(sqrt(c)*x - sqrt(c*x
^2 + a))^7*a^4*c^4*d^2*e^9 + 12120*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^5*d^4*e^7 - 285*(sqrt(c)*x - sqrt(c*x
^2 + a))^8*a^4*c^(7/2)*d*e^10 + 11640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(9/2)*d^3*e^8 + 4*a^6*c^(11/2)*d^5
*e^6 + 235*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 + 7020*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^9
- 336*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d^4*e^7 + 810*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*e^10
- 4038*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*d^3*e^8 + 390*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11
- 2330*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d^2*e^9 - 930*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*e^1
0 + 28*a^7*c^(9/2)*d^3*e^8 + 390*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*e^11 + 882*(sqrt(c)*x - sqrt(c*x^2 +
a))*a^7*c^4*d^2*e^9 + 321*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(7/2)*d*e^10 + 235*(sqrt(c)*x - sqrt(c*x^2 + a
))^3*a^7*c^3*e^11 - 81*a^8*c^(7/2)*d*e^10 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^3*e^11)/((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)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c
)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)
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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 )}^{3/2}}{{\left (d+e\,x\right )}^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + c*x^2)^(3/2)/(d + e*x)^7,x)
[Out]
int((a + c*x^2)^(3/2)/(d + e*x)^7, x)
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