\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^5} \, dx\) [553]

Optimal result

Integrand size = 19, antiderivative size = 287 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {5 c^2 \left (8 d \left (c d^2+a e^2\right )+e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {5 c^{5/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^6}-\frac {5 c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 e^6 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

-5/24*c*(d*(a*e^2+4*c*d^2)+3*e*(a*e^2+2*c*d^2)*x)*(c*x^2+a)^(3/2)/e^3/(a*e^2+c*d^2)/(e*x+d)^3-1/4*(c*x^2+a)^(5
/2)/e/(e*x+d)^4-5*c^(5/2)*d*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/e^6-5/8*c^2*(3*a^2*e^4+12*a*c*d^2*e^2+8*c^2*d^4
)*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)^(3/2)+5/8*c^2*(8*d*(a*e^2+c*d^2)
+e*(3*a*e^2+4*c*d^2)*x)*(c*x^2+a)^(1/2)/e^5/(a*e^2+c*d^2)/(e*x+d)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {747, 825, 827, 858, 223, 212, 739} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=-\frac {5 c^2 \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{3/2}}-\frac {5 c^{5/2} d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^6}+\frac {5 c^2 \sqrt {a+c x^2} \left (e x \left (3 a e^2+4 c d^2\right )+8 d \left (a e^2+c d^2\right )\right )}{8 e^5 (d+e x) \left (a e^2+c d^2\right )}-\frac {5 c \left (a+c x^2\right )^{3/2} \left (3 e x \left (a e^2+2 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{24 e^3 (d+e x)^3 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^5,x]

[Out]

(5*c^2*(8*d*(c*d^2 + a*e^2) + e*(4*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5*(c*d^2 + a*e^2)*(d + e*x)) - (5
*c*(d*(4*c*d^2 + a*e^2) + 3*e*(2*c*d^2 + a*e^2)*x)*(a + c*x^2)^(3/2))/(24*e^3*(c*d^2 + a*e^2)*(d + e*x)^3) - (
a + c*x^2)^(5/2)/(4*e*(d + e*x)^4) - (5*c^(5/2)*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^6 - (5*c^2*(8*c^2*d^
4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*e^6*(c*d^2 +
a*e^2)^(3/2))

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 825

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^
(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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}}{4 e (d+e x)^4}+\frac {(5 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx}{4 e} \\ & = -\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {(5 c) \int \frac {\left (4 a c d e-2 c \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{(d+e x)^2} \, dx}{16 e^3 \left (c d^2+a e^2\right )} \\ & = \frac {5 c^2 \left (8 d \left (c d^2+a e^2\right )+e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}+\frac {(5 c) \int \frac {4 a c e \left (4 c d^2+3 a e^2\right )-32 c^2 d \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{32 e^5 \left (c d^2+a e^2\right )} \\ & = \frac {5 c^2 \left (8 d \left (c d^2+a e^2\right )+e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {\left (5 c^3 d\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{e^6}+\frac {\left (5 c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 e^6 \left (c d^2+a e^2\right )} \\ & = \frac {5 c^2 \left (8 d \left (c d^2+a e^2\right )+e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {\left (5 c^3 d\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{e^6}-\frac {\left (5 c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\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 )}{8 e^6 \left (c d^2+a e^2\right )} \\ & = \frac {5 c^2 \left (8 d \left (c d^2+a e^2\right )+e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {5 c^{5/2} d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^6}-\frac {5 c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 e^6 \left (c d^2+a e^2\right )^{3/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(946\) vs. \(2(287)=574\).

Time = 10.11 (sec) , antiderivative size = 946, normalized size of antiderivative = 3.30 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {\frac {a e \left (-6 a^5 e^6+240 c^{9/2} d^3 x^3 \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right ) \left (\sqrt {c} x-\sqrt {a+c x^2}\right )-a^4 \left (-30 \sqrt {c} e^6 x \sqrt {a+c x^2}+c e^4 \left (11 d^2+20 d e x+105 e^2 x^2\right )\right )+5 a^3 \left (c^{3/2} e^4 x \sqrt {a+c x^2} \left (11 d^2+20 d e x+51 e^2 x^2\right )+c^2 e^2 \left (20 d^4+71 d^3 e x+61 d^2 e^2 x^2-5 d e^3 x^3-99 e^4 x^4\right )\right )+5 a^2 \left (c^{5/2} e \sqrt {a+c x^2} \left (-3 d^5-88 d^4 e x-289 d^3 e^2 x^2-312 d^2 e^3 x^3-108 d e^4 x^4+108 e^5 x^5\right )+c^3 \left (24 d^6+99 d^5 e x+304 d^4 e^2 x^2+643 d^3 e^3 x^3+648 d^2 e^4 x^4+264 d e^5 x^5-132 e^6 x^6\right )\right )+60 a \left (c^4 x^2 \left (16 d^6+61 d^5 e x+96 d^4 e^2 x^2+89 d^3 e^3 x^3+56 d^2 e^4 x^4+26 d e^5 x^5-4 e^6 x^6\right )+c^{7/2} x \sqrt {a+c x^2} \left (-8 d^6-31 d^5 e x-56 d^4 e^2 x^2-69 d^3 e^3 x^3-56 d^2 e^4 x^4-26 d e^5 x^5+4 e^6 x^6\right )\right )\right )}{\left (c d^2+a e^2\right ) (d+e x)^4 \left (a^2 \left (-5 \sqrt {c} x+\sqrt {a+c x^2}\right )+16 c^2 x^4 \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )+4 a c x^2 \left (-5 \sqrt {c} x+3 \sqrt {a+c x^2}\right )\right )}+\frac {240 c^4 d^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {360 a c^3 d^2 e^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {90 a^2 c^2 e^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 c^{5/2} \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5+60 d (d+e x)^4 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )\right )}{(d+e x)^4}}{24 e^6} \]

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^5,x]

[Out]

((a*e*(-6*a^5*e^6 + 240*c^(9/2)*d^3*x^3*(4*d^3 + 15*d^2*e*x + 20*d*e^2*x^2 + 10*e^3*x^3)*(Sqrt[c]*x - Sqrt[a +
c*x^2]) - a^4*(-30*Sqrt[c]*e^6*x*Sqrt[a + c*x^2] + c*e^4*(11*d^2 + 20*d*e*x + 105*e^2*x^2)) + 5*a^3*(c^(3/2)*
e^4*x*Sqrt[a + c*x^2]*(11*d^2 + 20*d*e*x + 51*e^2*x^2) + c^2*e^2*(20*d^4 + 71*d^3*e*x + 61*d^2*e^2*x^2 - 5*d*e
^3*x^3 - 99*e^4*x^4)) + 5*a^2*(c^(5/2)*e*Sqrt[a + c*x^2]*(-3*d^5 - 88*d^4*e*x - 289*d^3*e^2*x^2 - 312*d^2*e^3*
x^3 - 108*d*e^4*x^4 + 108*e^5*x^5) + c^3*(24*d^6 + 99*d^5*e*x + 304*d^4*e^2*x^2 + 643*d^3*e^3*x^3 + 648*d^2*e^
4*x^4 + 264*d*e^5*x^5 - 132*e^6*x^6)) + 60*a*(c^4*x^2*(16*d^6 + 61*d^5*e*x + 96*d^4*e^2*x^2 + 89*d^3*e^3*x^3 +
56*d^2*e^4*x^4 + 26*d*e^5*x^5 - 4*e^6*x^6) + c^(7/2)*x*Sqrt[a + c*x^2]*(-8*d^6 - 31*d^5*e*x - 56*d^4*e^2*x^2
- 69*d^3*e^3*x^3 - 56*d^2*e^4*x^4 - 26*d*e^5*x^5 + 4*e^6*x^6))))/((c*d^2 + a*e^2)*(d + e*x)^4*(a^2*(-5*Sqrt[c]
*x + Sqrt[a + c*x^2]) + 16*c^2*x^4*(-(Sqrt[c]*x) + Sqrt[a + c*x^2]) + 4*a*c*x^2*(-5*Sqrt[c]*x + 3*Sqrt[a + c*x
^2]))) + (240*c^4*d^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e^
2)^(3/2) + (360*a*c^3*d^2*e^2*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2
) - a*e^2)^(3/2) + (90*a^2*c^2*e^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(
c*d^2) - a*e^2)^(3/2) + (2*c^(5/2)*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 1
2*e^5*x^5 + 60*d*(d + e*x)^4*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]]))/(d + e*x)^4)/(24*e^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3030\) vs. \(2(261)=522\).

Time = 2.35 (sec) , antiderivative size = 3031, normalized size of antiderivative = 10.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(3031\)
default	\(\text {Expression too large to display}\)	\(6003\)

[In]

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

[Out]

c^2/e^5*(c*x^2+a)^(1/2)-1/e^5*(5*c^(5/2)*d/e*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+3*c^2*(a*e^2+5*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*c^2*d/e^3*(3*a*e^2+5*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))
)-3*c*(a^2*e^4+6*a*c*d^2*e^2+5*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)))+6*c*d*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^5*(-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))))+1/e^6*(-a^3*e^6-3*a^2*c*d^2*e^4-3*a*c^2*d^4*e^2-
c^3*d^6)*(-1/4/(a*e^2+c*d^2)*e^2/(x+d/e)^4*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+7/4*c*d*e/(a*
e^2+c*d^2)*(-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))))-3/4*c/(a*e^2+c*d^2)*e
^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)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (262) = 524\).

Time = 7.33 (sec) , antiderivative size = 3733, normalized size of antiderivative = 13.01 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/48*(120*(c^4*d^9 + 2*a*c^3*d^7*e^2 + a^2*c^2*d^5*e^4 + (c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8)*x^4
+ 4*(c^4*d^6*e^3 + 2*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^3 + 6*(c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^2*c^2*d^3*e^6
)*x^2 + 4*(c^4*d^8*e + 2*a*c^3*d^6*e^3 + a^2*c^2*d^4*e^5)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*
x - a) + 15*(8*c^4*d^8 + 12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^3*d^2*e^6 + 3*a^2*c^2*
e^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 +
3*a^2*c^2*d^2*e^6)*x^2 + 4*(8*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*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*(120*c^4*d^8*e + 220*a*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 - 17*a^3*c*d^2*e^7
- 6*a^4*e^9 + 24*(c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x^4 + 5*(50*c^4*d^5*e^4 + 97*a*c^3*d^3*e^6 + 4
7*a^2*c^2*d*e^8)*x^3 + (520*c^4*d^6*e^3 + 968*a*c^3*d^4*e^5 + 421*a^2*c^2*d^2*e^7 - 27*a^3*c*e^9)*x^2 + 5*(84*
c^4*d^7*e^2 + 155*a*c^3*d^5*e^4 + 67*a^2*c^2*d^3*e^6 - 4*a^3*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^2*d^8*e^6 + 2*a*c
*d^6*e^8 + a^2*d^4*e^10 + (c^2*d^4*e^10 + 2*a*c*d^2*e^12 + a^2*e^14)*x^4 + 4*(c^2*d^5*e^9 + 2*a*c*d^3*e^11 + a
^2*d*e^13)*x^3 + 6*(c^2*d^6*e^8 + 2*a*c*d^4*e^10 + a^2*d^2*e^12)*x^2 + 4*(c^2*d^7*e^7 + 2*a*c*d^5*e^9 + a^2*d^
3*e^11)*x), 1/48*(240*(c^4*d^9 + 2*a*c^3*d^7*e^2 + a^2*c^2*d^5*e^4 + (c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*
d*e^8)*x^4 + 4*(c^4*d^6*e^3 + 2*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^3 + 6*(c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^2*
c^2*d^3*e^6)*x^2 + 4*(c^4*d^8*e + 2*a*c^3*d^6*e^3 + a^2*c^2*d^4*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2
+ a)) + 15*(8*c^4*d^8 + 12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^3*d^2*e^6 + 3*a^2*c^2*e
^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 + 3
*a^2*c^2*d^2*e^6)*x^2 + 4*(8*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*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*(120*c^4*d^8*e + 220*a*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 - 17*a^3*c*d^2*e^7
- 6*a^4*e^9 + 24*(c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x^4 + 5*(50*c^4*d^5*e^4 + 97*a*c^3*d^3*e^6 + 47
*a^2*c^2*d*e^8)*x^3 + (520*c^4*d^6*e^3 + 968*a*c^3*d^4*e^5 + 421*a^2*c^2*d^2*e^7 - 27*a^3*c*e^9)*x^2 + 5*(84*c
^4*d^7*e^2 + 155*a*c^3*d^5*e^4 + 67*a^2*c^2*d^3*e^6 - 4*a^3*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^2*d^8*e^6 + 2*a*c*
d^6*e^8 + a^2*d^4*e^10 + (c^2*d^4*e^10 + 2*a*c*d^2*e^12 + a^2*e^14)*x^4 + 4*(c^2*d^5*e^9 + 2*a*c*d^3*e^11 + a^
2*d*e^13)*x^3 + 6*(c^2*d^6*e^8 + 2*a*c*d^4*e^10 + a^2*d^2*e^12)*x^2 + 4*(c^2*d^7*e^7 + 2*a*c*d^5*e^9 + a^2*d^3
*e^11)*x), -1/24*(15*(8*c^4*d^8 + 12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^3*d^2*e^6 + 3
*a^2*c^2*e^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d
^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^2 + 4*(8*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*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)) -
60*(c^4*d^9 + 2*a*c^3*d^7*e^2 + a^2*c^2*d^5*e^4 + (c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8)*x^4 + 4*(c^4
*d^6*e^3 + 2*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^3 + 6*(c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^2*c^2*d^3*e^6)*x^2 +
4*(c^4*d^8*e + 2*a*c^3*d^6*e^3 + a^2*c^2*d^4*e^5)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) -
(120*c^4*d^8*e + 220*a*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 - 17*a^3*c*d^2*e^7 - 6*a^4*e^9 + 24*(c^4*d^4*e^5 + 2*
a*c^3*d^2*e^7 + a^2*c^2*e^9)*x^4 + 5*(50*c^4*d^5*e^4 + 97*a*c^3*d^3*e^6 + 47*a^2*c^2*d*e^8)*x^3 + (520*c^4*d^6
*e^3 + 968*a*c^3*d^4*e^5 + 421*a^2*c^2*d^2*e^7 - 27*a^3*c*e^9)*x^2 + 5*(84*c^4*d^7*e^2 + 155*a*c^3*d^5*e^4 + 6
7*a^2*c^2*d^3*e^6 - 4*a^3*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^2*d^8*e^6 + 2*a*c*d^6*e^8 + a^2*d^4*e^10 + (c^2*d^4*
e^10 + 2*a*c*d^2*e^12 + a^2*e^14)*x^4 + 4*(c^2*d^5*e^9 + 2*a*c*d^3*e^11 + a^2*d*e^13)*x^3 + 6*(c^2*d^6*e^8 + 2
*a*c*d^4*e^10 + a^2*d^2*e^12)*x^2 + 4*(c^2*d^7*e^7 + 2*a*c*d^5*e^9 + a^2*d^3*e^11)*x), -1/24*(15*(8*c^4*d^8 +
12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^3*d^2*e^6 + 3*a^2*c^2*e^8)*x^4 + 4*(8*c^4*d^5*e
^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^2 +
4*(8*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*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)) - 120*(c^4*d^9 + 2*a*c^3*d^7*e^2 + a
^2*c^2*d^5*e^4 + (c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8)*x^4 + 4*(c^4*d^6*e^3 + 2*a*c^3*d^4*e^5 + a^2*
c^2*d^2*e^7)*x^3 + 6*(c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^2*c^2*d^3*e^6)*x^2 + 4*(c^4*d^8*e + 2*a*c^3*d^6*e^3 +
a^2*c^2*d^4*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (120*c^4*d^8*e + 220*a*c^3*d^6*e^3 + 89*a^2*
c^2*d^4*e^5 - 17*a^3*c*d^2*e^7 - 6*a^4*e^9 + 24*(c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x^4 + 5*(50*c^4*
d^5*e^4 + 97*a*c^3*d^3*e^6 + 47*a^2*c^2*d*e^8)*x^3 + (520*c^4*d^6*e^3 + 968*a*c^3*d^4*e^5 + 421*a^2*c^2*d^2*e^
7 - 27*a^3*c*e^9)*x^2 + 5*(84*c^4*d^7*e^2 + 155*a*c^3*d^5*e^4 + 67*a^2*c^2*d^3*e^6 - 4*a^3*c*d*e^8)*x)*sqrt(c*
x^2 + a))/(c^2*d^8*e^6 + 2*a*c*d^6*e^8 + a^2*d^4*e^10 + (c^2*d^4*e^10 + 2*a*c*d^2*e^12 + a^2*e^14)*x^4 + 4*(c^
2*d^5*e^9 + 2*a*c*d^3*e^11 + a^2*d*e^13)*x^3 + 6*(c^2*d^6*e^8 + 2*a*c*d^4*e^10 + a^2*d^2*e^12)*x^2 + 4*(c^2*d^
7*e^7 + 2*a*c*d^5*e^9 + a^2*d^3*e^11)*x)]

Sympy [F]

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

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**5,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**5, x)

Maxima [F(-2)]

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

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^5,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 [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^5,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^5, x)