\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^4} \, dx\) [552]

Optimal result

Integrand size = 19, antiderivative size = 222 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {5 c \left (4 c d^2+a e^2+2 c d e x\right ) \sqrt {a+c x^2}}{2 e^5 (d+e x)}+\frac {5 c (2 d+e x) \left (a+c x^2\right )^{3/2}}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {5 c^{3/2} \left (4 c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 e^6}+\frac {5 c^2 d \left (4 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^6 \sqrt {c d^2+a e^2}} \]

[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)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {747, 827, 858, 223, 212, 739} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {5 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+4 c d^2\right )}{2 e^6}+\frac {5 c^2 d \left (3 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^6 \sqrt {a e^2+c d^2}}-\frac {5 c \sqrt {a+c x^2} \left (a e^2+4 c d^2+2 c d e x\right )}{2 e^5 (d+e x)}+\frac {5 c \left (a+c x^2\right )^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac {\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 2.66 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {\frac {e \sqrt {a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+15 d e x+14 e^2 x^2\right )+c^2 \left (60 d^4+150 d^3 e x+110 d^2 e^2 x^2+15 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^3}-\frac {30 c^2 d \left (4 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+15 c^{3/2} \left (4 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 e^6} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1726\) vs. \(2(194)=388\).

Time = 2.15 (sec) , antiderivative size = 1727, normalized size of antiderivative = 7.78

method	result	size
risch	\(\text {Expression too large to display}\)	\(1727\)
default	\(\text {Expression too large to display}\)	\(3656\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/2*c^2*(-e*x+8*d)*(c*x^2+a)^(1/2)/e^5+1/2/e^5*(5*c^(3/2)*(a*e^2+4*c*d^2)/e*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+8*c
^2*d/e^2*(3*a*e^2+5*c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/
e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+6*c*(a^2*e^4+6*a*c*d^2*e^2+5*c^2*d^
4)/e^3*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c*d*e/(a*e^2+c*d^2)
/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*
c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))-12*c*d*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^4*(-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)^(1/2)+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)^(1/2)-c*d*e/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(
1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c
*d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(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)))+1/e^5*(2*
a^3*e^6+6*a^2*c*d^2*e^4+6*a*c^2*d^4*e^2+2*c^3*d^6)*(-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)^(1/2)+5/3*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)^(1/2)+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)^(1/2)-c*d*e/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+
d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(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)))-2/3*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)^(1/2)-c*d*e/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2
*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)
^(1/2))/(x+d/e)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (195) = 390\).

Time = 2.26 (sec) , antiderivative size = 2501, normalized size of antiderivative = 11.27 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/12*(15*(4*c^3*d^7 + 5*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^3*d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3
*(4*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^3*d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sq
rt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a
*c^2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*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*(60*c^3*d^6*e + 65*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e
^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 15*(c^3*d^3*e^4 + a*c^2*d*e^6)*x^3 + 2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e
^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*d^5*e^2 + 11*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 +
a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x), 1/12*(30*(
4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 +
3*(4*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)) + 15*(4*c^3*d^7 + 5*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^
3*d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(4*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^
3*d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(60*
c^3*d^6*e + 65*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 15*(c^3*d^3*e^4
+ a*c^2*d*e^6)*x^3 + 2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*d^5*e^2 + 11*a*c^2*
d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^3 + 3*(c*d^3*e^8 +
a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x), -1/12*(30*(4*c^3*d^7 + 5*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^3*
d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(4*c^3*d^5*e^2 + 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^3*
d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 15*(4*c^3*d^6 + 3*a*
c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*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*(60*c^3*d^6*e + 65*a*c^2*d
^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 15*(c^3*d^3*e^4 + a*c^2*d*e^6)*x^3 +
2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*d^5*e^2 + 11*a*c^2*d^3*e^4 + a^2*c*d*e^6)
*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d
^4*e^7 + a*d^2*e^9)*x), 1/6*(15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c^3*
d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*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)) - 15*(4*c^3*d^7 + 5*a*c^
2*d^5*e^2 + a^2*c*d^3*e^4 + (4*c^3*d^4*e^3 + 5*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(4*c^3*d^5*e^2 + 5*a*c^2*d^3
*e^4 + a^2*c*d*e^6)*x^2 + 3*(4*c^3*d^6*e + 5*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt
(c*x^2 + a)) - (60*c^3*d^6*e + 65*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 - 3*(c^3*d^2*e^5 + a*c^2*e^7)*x^
4 + 15*(c^3*d^3*e^4 + a*c^2*d*e^6)*x^3 + 2*(55*c^3*d^4*e^3 + 62*a*c^2*d^2*e^5 + 7*a^2*c*e^7)*x^2 + 15*(10*c^3*
d^5*e^2 + 11*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c*d^5*e^6 + a*d^3*e^8 + (c*d^2*e^9 + a*e^11)*x^
3 + 3*(c*d^3*e^8 + a*d*e^10)*x^2 + 3*(c*d^4*e^7 + a*d^2*e^9)*x)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (195) = 390\).

Time = 0.65 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.66 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {1}{2} \, \sqrt {c x^{2} + a} {\left (\frac {c^{2} x}{e^{4}} - \frac {8 \, c^{2} d}{e^{5}}\right )} - \frac {5 \, {\left (4 \, c^{\frac {5}{2}} d^{2} + a c^{\frac {3}{2}} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, e^{6}} - \frac {5 \, {\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} e^{6}} - \frac {60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{2} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} - 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} + 188 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} - 226 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} - 354 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{5} + 222 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} + 57 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} - 47 \, a^{3} c^{\frac {5}{2}} d^{2} e^{3} - 14 \, a^{4} c^{\frac {3}{2}} e^{5}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3} e^{6}} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/2*sqrt(c*x^2 + a)*(c^2*x/e^4 - 8*c^2*d/e^5) - 5/2*(4*c^(5/2)*d^2 + a*c^(3/2)*e^2)*log(abs(-sqrt(c)*x + sqrt(
c*x^2 + a)))/e^6 - 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))/(sqrt(-c*d^2 - a*e^2)*e^6) - 1/3*(60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*d^3*e^2 + 27*(sqrt(c)*
x - sqrt(c*x^2 + a))^5*a*c^2*d*e^4 + 210*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e + 27*(sqrt(c)*x - sqrt(
c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 - 18*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*e^5 + 188*(sqrt(c)*x - sqrt(c
*x^2 + a))^3*c^4*d^5 - 226*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d^3*e^2 - 84*(sqrt(c)*x - sqrt(c*x^2 + a))^3*
a^2*c^2*d*e^4 - 354*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c
^(3/2)*e^5 + 222*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d^3*e^2 + 57*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^
4 - 47*a^3*c^(5/2)*d^2*e^3 - 14*a^4*c^(3/2)*e^5)/(((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*e^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]

[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)