3.6.52 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^4} \, dx\) [552]
Optimal. Leaf size=222 \[ -\frac {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {5 c^{3/2} \left (4 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (4 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^6 \sqrt {c d^2+a e^2}} \]
[Out]
5/6*c*(e*x+2*d)*(c*x^2+a)^(3/2)/e^3/(e*x+d)^2-1/3*(c*x^2+a)^(5/2)/e/(e*x+d)^3+5/2*c^(3/2)*(a*e^2+4*c*d^2)*arct
anh(x*c^(1/2)/(c*x^2+a)^(1/2))/e^6+5/2*c^2*d*(3*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^6/(a*e^2+c*d^2)^(1/2)-5/2*c*(2*c*d*e*x+a*e^2+4*c*d^2)*(c*x^2+a)^(1/2)/e^5/(e*x+d)
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Rubi [A]
time = 0.15, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {747, 827, 858,
223, 212, 739} \begin {gather*} \frac {5 c^{3/2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (3 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 )}{2 e^6 \sqrt {a e^2+c d^2}}-\frac {5 c \sqrt {a+c x^2} \left (a e^2+4 c d^2+2 c d e x\right )}{2 e^5 (d+e x)}+\frac {5 c \left (a+c x^2\right )^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + c*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
(-5*c*(4*c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a + c*x^2])/(2*e^5*(d + e*x)) + (5*c*(2*d + e*x)*(a + c*x^2)^(3/2))/(
6*e^3*(d + e*x)^2) - (a + c*x^2)^(5/2)/(3*e*(d + e*x)^3) + (5*c^(3/2)*(4*c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sq
rt[a + c*x^2]])/(2*e^6) + (5*c^2*d*(4*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*e^6*Sqrt[c*d^2 + a*e^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 747
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 1))), x] - Dist[2*c*(p/(e*(m + 1))), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 827
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx &=-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {(5 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^3} \, dx}{3 e}\\ &=\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {(5 c) \int \frac {(-4 a e+8 c d x) \sqrt {a+c x^2}}{(d+e x)^2} \, dx}{8 e^3}\\ &=-\frac {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {(5 c) \int \frac {-16 a c d e+8 c \left (4 c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 e^5}\\ &=-\frac {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {\left (5 c^2 \left (4 c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 e^6}-\frac {\left (5 c^2 d \left (4 c d^2+3 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 e^6}\\ &=-\frac {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {\left (5 c^2 \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 )}{2 e^6}+\frac {\left (5 c^2 d \left (4 c d^2+3 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 )}{2 e^6}\\ &=-\frac {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {5 c^{3/2} \left (4 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (4 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 e^6 \sqrt {c d^2+a e^2}}\\ \end {align*}
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Mathematica [A]
time = 1.88, size = 228, normalized size = 1.03 \begin {gather*} -\frac {\frac {e \sqrt {a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+15 d e x+14 e^2 x^2\right )+c^2 \left (60 d^4+150 d^3 e x+110 d^2 e^2 x^2+15 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^3}-\frac {30 c^2 d \left (4 c d^2+3 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 )}{\sqrt {-c d^2-a e^2}}+15 c^{3/2} \left (4 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + c*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
-1/6*((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 15*d*e*x + 14*e^2*x^2) + c^2*(60*d^4 + 150*d^3*e*x + 11
0*d^2*e^2*x^2 + 15*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x)^3 - (30*c^2*d*(4*c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*(d + e
*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/Sqrt[-(c*d^2) - a*e^2] + 15*c^(3/2)*(4*c*d^2 + a*e^2)*Log[-(
Sqrt[c]*x) + Sqrt[a + c*x^2]])/e^6
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3655\) vs.
\(2(194)=388\).
time = 0.50, size = 3656, normalized size = 16.47
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(3627\) |
| default |
\(\text {Expression too large to display}\) |
\(3656\) |
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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)^4,x,method=_RETURNVERBOSE)
[Out]
1/e^4*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(7/2)-1/3*c*d*e/(a*e^2
+c*d^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(7/2)-3/2*c*d*e/(a*e
^2+c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(7/2)-5*c*d*e/(a*e^2+c
*d^2)*(1/5*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-c*d/e*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)
^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c
*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2
)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))))+(a*e^2+c*d^2)/e^2*(1/
3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/
e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(
1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(
a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e
^2)^(1/2))-(a*e^2+c*d^2)/e^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)))))+6*c/(a*e^2+c*d^2)*e^2*(1/12*(2
*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)+5/24*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*
d^2/e^2)/c*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2
+c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2
)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)
+(a*e^2+c*d^2)/e^2)^(1/2))))))+5/2*c/(a*e^2+c*d^2)*e^2*(1/5*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5
/2)-c*d/e*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+
c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)
+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+
(a*e^2+c*d^2)/e^2)^(1/2))))+(a*e^2+c*d^2)/e^2*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e
*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2
-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+
(a*e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(
1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^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))))))+4/3*c/(a*e^2+c*d^2)*e^2*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+
c*d^2)/e^2)^(7/2)-5*c*d*e/(a*e^2+c*d^2)*(1/5*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-c*d/e*(1/8*
(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+c*d^2)/e^2-4*c^
2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^
2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e
^2)^(1/2))))+(a*e^2+c*d^2)/e^2*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c*(x+d
/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)
/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d^2)/e
^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)
^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^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))
)))+6*c/(a*e^2+c*d^2)*e^2*(1/12*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)+
5/24*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+
c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d
/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^
(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)))))))
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 978 vs.
\(2 (191) = 382\).
time = 0.41, size = 978, normalized size = 4.41 \begin {gather*} -\frac {5 \, c^{5} d^{6} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{4 \, {\left (c^{\frac {5}{2}} d^{4} e^{6} + 2 \, a c^{\frac {3}{2}} d^{2} e^{8} + a^{2} \sqrt {c} e^{10}\right )}} - \frac {5 \, a c^{4} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{4 \, {\left (c^{\frac {5}{2}} d^{4} e^{4} + 2 \, a c^{\frac {3}{2}} d^{2} e^{6} + a^{2} \sqrt {c} e^{8}\right )}} + \frac {5 \, \sqrt {c x^{2} + a} c^{4} d^{4} x}{4 \, {\left (c^{2} d^{4} e^{4} + 2 \, a c d^{2} e^{6} + a^{2} e^{8}\right )}} - \frac {5 \, c^{3} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-7\right )}}{2 \, \sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d^{3}}{6 \, {\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d^{2} x}{6 \, {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )}} + \frac {5 \, \sqrt {c x^{2} + a} a c^{3} d^{2} x}{4 \, {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )}} + \frac {45}{4} \, c^{\frac {5}{2}} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-6\right )} - \frac {5 \, a c^{3} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{4 \, {\left (c^{\frac {3}{2}} d^{2} e^{4} + a \sqrt {c} e^{6}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{2} d^{2}}{2 \, {\left (c^{2} d^{4} x e^{2} + c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 2 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )}} - \frac {5 \, \sqrt {c x^{2} + a} c^{3} d^{3}}{2 \, {\left (c d^{2} e^{5} + a e^{7}\right )}} + \frac {15 \, \sqrt {c x^{2} + a} c^{3} d^{2} x}{4 \, {\left (c d^{2} e^{4} + a e^{6}\right )}} - \frac {15}{2} \, \sqrt {c d^{2} e^{\left (-2\right )} + a} c^{2} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-5\right )} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} c d}{6 \, {\left (c^{2} d^{4} x^{2} e + c^{2} d^{6} e^{\left (-1\right )} + 2 \, c^{2} d^{5} x + 2 \, a c d^{2} x^{2} e^{3} + 4 \, a c d^{3} x e^{2} + 2 \, a c d^{4} e + a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4} + a^{2} d^{2} e^{3}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{2} d}{6 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac {5}{2} \, a c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )} - \frac {15}{2} \, \sqrt {c x^{2} + a} c^{2} d e^{\left (-5\right )} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d}{2 \, {\left (c d^{2} e^{3} + a e^{5}\right )}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} x}{3 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} + \frac {5 \, \sqrt {c x^{2} + a} a c^{2} x}{2 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}}}{3 \, {\left (c d^{2} x^{3} e^{2} + 3 \, c d^{3} x^{2} e + c d^{5} e^{\left (-1\right )} + 3 \, c d^{4} x + a x^{3} e^{4} + 3 \, a d x^{2} e^{3} + 3 \, a d^{2} x e^{2} + a d^{3} e\right )}} - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} c}{3 \, {\left (c d^{2} x e^{2} + c d^{3} e + a x e^{4} + a d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="maxima")
[Out]
-5/4*c^5*d^6*arcsinh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^6 + 2*a*c^(3/2)*d^2*e^8 + a^2*sqrt(c)*e^10) - 5/4*a*c^4*d^4
*arcsinh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^4 + 2*a*c^(3/2)*d^2*e^6 + a^2*sqrt(c)*e^8) + 5/4*sqrt(c*x^2 + a)*c^4*d^
4*x/(c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2*e^8) - 5/2*c^3*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a
*c)*abs(x*e + d)))*e^(-7)/sqrt(c*d^2*e^(-2) + a) - 5/6*(c*x^2 + a)^(3/2)*c^3*d^3/(c^2*d^4*e^3 + 2*a*c*d^2*e^5
+ a^2*e^7) + 5/6*(c*x^2 + a)^(3/2)*c^3*d^2*x/(c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) + 5/4*sqrt(c*x^2 + a)*a*c
^3*d^2*x/(c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) + 45/4*c^(5/2)*d^2*arcsinh(c*x/sqrt(a*c))*e^(-6) - 5/4*a*c^3*
d^2*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^4 + a*sqrt(c)*e^6) - 1/2*(c*x^2 + a)^(5/2)*c^2*d^2/(c^2*d^4*x*e^2 +
c^2*d^5*e + 2*a*c*d^2*x*e^4 + 2*a*c*d^3*e^3 + a^2*x*e^6 + a^2*d*e^5) - 5/2*sqrt(c*x^2 + a)*c^3*d^3/(c*d^2*e^5
+ a*e^7) + 15/4*sqrt(c*x^2 + a)*c^3*d^2*x/(c*d^2*e^4 + a*e^6) - 15/2*sqrt(c*d^2*e^(-2) + a)*c^2*d*arcsinh(c*d*
x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-5) + 1/6*(c*x^2 + a)^(7/2)*c*d/(c^2*d^4*x^2*e +
c^2*d^6*e^(-1) + 2*c^2*d^5*x + 2*a*c*d^2*x^2*e^3 + 4*a*c*d^3*x*e^2 + 2*a*c*d^4*e + a^2*x^2*e^5 + 2*a^2*d*x*e^
4 + a^2*d^2*e^3) - 1/6*(c*x^2 + a)^(5/2)*c^2*d/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + 5/2*a*c^(3/2)*arcsinh(c
*x/sqrt(a*c))*e^(-4) - 15/2*sqrt(c*x^2 + a)*c^2*d*e^(-5) - 5/2*(c*x^2 + a)^(3/2)*c^2*d/(c*d^2*e^3 + a*e^5) + 5
/3*(c*x^2 + a)^(3/2)*c^2*x/(c*d^2*e^2 + a*e^4) + 5/2*sqrt(c*x^2 + a)*a*c^2*x/(c*d^2*e^2 + a*e^4) - 1/3*(c*x^2
+ a)^(7/2)/(c*d^2*x^3*e^2 + 3*c*d^3*x^2*e + c*d^5*e^(-1) + 3*c*d^4*x + a*x^3*e^4 + 3*a*d*x^2*e^3 + 3*a*d^2*x*e
^2 + a*d^3*e) - 4/3*(c*x^2 + a)^(5/2)*c/(c*d^2*x*e^2 + c*d^3*e + a*x*e^4 + a*d*e^3)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 587 vs.
\(2 (191) = 382\).
time = 6.68, size = 2416, normalized size = 10.88 \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)^4,x, algorithm="fricas")
[Out]
[1/12*(15*(12*c^3*d^6*x*e + 4*c^3*d^7 + a^2*c*x^3*e^7 + 3*a^2*c*d*x^2*e^6 + (5*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*
e^5 + (15*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (4*c^3*d^4*x^3 + 15*a*c^2*d^4*x)*e^3 + (12*c^3*d^5*x^2 + 5*a*c^2*d^
5)*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 15*(12*c^3*d^5*x*e + 4*c^3*d^6 + 3*a*c^2*d*x
^3*e^5 + 9*a*c^2*d^2*x^2*e^4 + (4*c^3*d^3*x^3 + 9*a*c^2*d^3*x)*e^3 + 3*(4*c^3*d^4*x^2 + a*c^2*d^4)*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*(150*c^3*d^5*x*e^2 + 60*c^3*d^6*e - (3*a*c^2*x^4 - 1
4*a^2*c*x^2 - 2*a^3)*e^7 + 15*(a*c^2*d*x^3 + a^2*c*d*x)*e^6 - (3*c^3*d^2*x^4 - 124*a*c^2*d^2*x^2 - 7*a^2*c*d^2
)*e^5 + 15*(c^3*d^3*x^3 + 11*a*c^2*d^3*x)*e^4 + 5*(22*c^3*d^4*x^2 + 13*a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c*d
^4*x*e^7 + c*d^5*e^6 + a*x^3*e^11 + 3*a*d*x^2*e^10 + (c*d^2*x^3 + 3*a*d^2*x)*e^9 + (3*c*d^3*x^2 + a*d^3)*e^8),
-1/12*(30*(12*c^3*d^5*x*e + 4*c^3*d^6 + 3*a*c^2*d*x^3*e^5 + 9*a*c^2*d^2*x^2*e^4 + (4*c^3*d^3*x^3 + 9*a*c^2*d^
3*x)*e^3 + 3*(4*c^3*d^4*x^2 + a*c^2*d^4)*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)) - 15*(12*c^3*d^6*x*e + 4*c^3*d^7 + a^2*c*x^3*e^
7 + 3*a^2*c*d*x^2*e^6 + (5*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*e^5 + (15*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (4*c^3*d^
4*x^3 + 15*a*c^2*d^4*x)*e^3 + (12*c^3*d^5*x^2 + 5*a*c^2*d^5)*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqr
t(c)*x - a) + 2*(150*c^3*d^5*x*e^2 + 60*c^3*d^6*e - (3*a*c^2*x^4 - 14*a^2*c*x^2 - 2*a^3)*e^7 + 15*(a*c^2*d*x^3
+ a^2*c*d*x)*e^6 - (3*c^3*d^2*x^4 - 124*a*c^2*d^2*x^2 - 7*a^2*c*d^2)*e^5 + 15*(c^3*d^3*x^3 + 11*a*c^2*d^3*x)*
e^4 + 5*(22*c^3*d^4*x^2 + 13*a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c*d^4*x*e^7 + c*d^5*e^6 + a*x^3*e^11 + 3*a*d*
x^2*e^10 + (c*d^2*x^3 + 3*a*d^2*x)*e^9 + (3*c*d^3*x^2 + a*d^3)*e^8), -1/12*(30*(12*c^3*d^6*x*e + 4*c^3*d^7 + a
^2*c*x^3*e^7 + 3*a^2*c*d*x^2*e^6 + (5*a*c^2*d^2*x^3 + 3*a^2*c*d^2*x)*e^5 + (15*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4
+ (4*c^3*d^4*x^3 + 15*a*c^2*d^4*x)*e^3 + (12*c^3*d^5*x^2 + 5*a*c^2*d^5)*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c
*x^2 + a)) - 15*(12*c^3*d^5*x*e + 4*c^3*d^6 + 3*a*c^2*d*x^3*e^5 + 9*a*c^2*d^2*x^2*e^4 + (4*c^3*d^3*x^3 + 9*a*c
^2*d^3*x)*e^3 + 3*(4*c^3*d^4*x^2 + a*c^2*d^4)*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*(150*c^3*d^5*x*e^2 + 60*c^3*d^6*e - (3*a*c^2*x^4 - 14*a^2*c*x^2 - 2*a^3)*e^7 + 15*(a*c^2*d*x^3 + a^2*c*d*
x)*e^6 - (3*c^3*d^2*x^4 - 124*a*c^2*d^2*x^2 - 7*a^2*c*d^2)*e^5 + 15*(c^3*d^3*x^3 + 11*a*c^2*d^3*x)*e^4 + 5*(22
*c^3*d^4*x^2 + 13*a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c*d^4*x*e^7 + c*d^5*e^6 + a*x^3*e^11 + 3*a*d*x^2*e^10 +
(c*d^2*x^3 + 3*a*d^2*x)*e^9 + (3*c*d^3*x^2 + a*d^3)*e^8), -1/6*(15*(12*c^3*d^5*x*e + 4*c^3*d^6 + 3*a*c^2*d*x^3
*e^5 + 9*a*c^2*d^2*x^2*e^4 + (4*c^3*d^3*x^3 + 9*a*c^2*d^3*x)*e^3 + 3*(4*c^3*d^4*x^2 + a*c^2*d^4)*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)) + 15*(12*c^3*d^6*x*e + 4*c^3*d^7 + a^2*c*x^3*e^7 + 3*a^2*c*d*x^2*e^6 + (5*a*c^2*d^2*x^3 + 3*a^2*c*d^2
*x)*e^5 + (15*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + (4*c^3*d^4*x^3 + 15*a*c^2*d^4*x)*e^3 + (12*c^3*d^5*x^2 + 5*a*c^
2*d^5)*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (150*c^3*d^5*x*e^2 + 60*c^3*d^6*e - (3*a*c^2*x^4 - 1
4*a^2*c*x^2 - 2*a^3)*e^7 + 15*(a*c^2*d*x^3 + a^2*c*d*x)*e^6 - (3*c^3*d^2*x^4 - 124*a*c^2*d^2*x^2 - 7*a^2*c*d^2
)*e^5 + 15*(c^3*d^3*x^3 + 11*a*c^2*d^3*x)*e^4 + 5*(22*c^3*d^4*x^2 + 13*a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c*d
^4*x*e^7 + c*d^5*e^6 + a*x^3*e^11 + 3*a*d*x^2*e^10 + (c*d^2*x^3 + 3*a*d^2*x)*e^9 + (3*c*d^3*x^2 + a*d^3)*e^8)]
________________________________________________________________________________________
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 )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+a)**(5/2)/(e*x+d)**4,x)
[Out]
Integral((a + c*x**2)**(5/2)/(d + e*x)**4, x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 575 vs.
\(2 (191) = 382\).
time = 1.88, size = 575, normalized size = 2.59 \begin {gather*} -\frac {5}{2} \, {\left (4 \, c^{\frac {5}{2}} d^{2} + a c^{\frac {3}{2}} e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) - \frac {5 \, {\left (4 \, c^{3} d^{3} + 3 \, a c^{2} 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 ) e^{\left (-6\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {1}{2} \, {\left (c^{2} x e^{\left (-4\right )} - 8 \, c^{2} d e^{\left (-5\right )}\right )} \sqrt {c x^{2} + a} - \frac {{\left (210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 188 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{2} - 354 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e - 226 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 222 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} - 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} - 47 \, a^{3} c^{\frac {5}{2}} d^{2} e^{3} + 57 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{5} - 14 \, a^{4} c^{\frac {3}{2}} e^{5}\right )} e^{\left (-6\right )}}{3 \, {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="giac")
[Out]
-5/2*(4*c^(5/2)*d^2 + a*c^(3/2)*e^2)*e^(-6)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) - 5*(4*c^3*d^3 + 3*a*c^2*d*
e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))*e^(-6)/sqrt(-c*d^2 - a*e^2) +
1/2*(c^2*x*e^(-4) - 8*c^2*d*e^(-5))*sqrt(c*x^2 + a) - 1/3*(210*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e
+ 188*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d^5 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*d^3*e^2 - 354*(sqrt(c)*
x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e - 226*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d^3*e^2 + 27*(sqrt(c)*x - s
qrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 + 27*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*e^4 + 222*(sqrt(c)*x - sqrt(c
*x^2 + a))*a^2*c^3*d^3*e^2 - 84*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*e^4 - 18*(sqrt(c)*x - sqrt(c*x^2 + a
))^4*a^2*c^(3/2)*e^5 - 47*a^3*c^(5/2)*d^2*e^3 + 57*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^4 + 24*(sqrt(c)*x
- sqrt(c*x^2 + a))^2*a^3*c^(3/2)*e^5 - 14*a^4*c^(3/2)*e^5)*e^(-6)/((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqr
t(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^3
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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 )}^4} \,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)^4,x)
[Out]
int((a + c*x^2)^(5/2)/(d + e*x)^4, x)
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