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

Optimal. Leaf size=287 \[ \frac {5 c^2 \left (8 d \left (c d^2+a e^2\right )+e \left (4 c d^2+3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 c \left (d \left (4 c d^2+a e^2\right )+3 e \left (2 c d^2+a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 e^3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {5 c^{5/2} d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^6}-\frac {5 c^2 \left (8 c^2 d^4+12 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 e^6 \left (c d^2+a e^2\right )^{3/2}} \]

[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)

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Rubi [A]
time = 0.20, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {747, 825, 827, 858, 223, 212, 739} \begin {gather*} -\frac {5 c^2 \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\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 \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(946\) vs. \(2(287)=574\).
time = 6.41, size = 946, normalized size = 3.30 \begin {gather*} \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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6002\) vs. \(2(261)=522\).
time = 0.50, size = 6003, normalized size = 20.92

method	result	size
risch	\(\text {Expression too large to display}\)	\(5176\)
default	\(\text {Expression too large to display}\)	\(6003\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^5,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. 1813 vs. \(2 (253) = 506\).
time = 0.45, size = 1813, normalized size = 6.32 \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)^5,x, algorithm="maxima")

[Out]

-5/16*c^6*d^7*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*sqr
t(c)*e^12) - 5/16*a*c^5*d^5*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e^4 + 3*a*c^(5/2)*d^4*e^6 + 3*a^2*c^(3/2)*d^2*
e^8 + a^3*sqrt(c)*e^10) + 5/16*sqrt(c*x^2 + a)*c^5*d^5*x/(c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 + a^
3*e^10) + 45/16*c^5*d^5*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/
24*(c*x^2 + a)^(3/2)*c^4*d^4/(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) + 5/24*(c*x^2 + a)^(3
/2)*c^4*d^3*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) + 5/16*sqrt(c*x^2 + a)*a*c^4*d^3*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) - 5/8*c^4*d^4*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)^(3/2) + 5/2*a*c^4*d^3*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) - 1/8*(c*x^2 + a)^(5/2)*c^3*d^3/(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) - 5/8*s
qrt(c*x^2 + a)*c^4*d^4/(c^2*d^4*e^5 + 2*a*c*d^2*e^7 + a^2*e^9) - 15/8*sqrt(c*x^2 + a)*c^4*d^3*x/(c^2*d^4*e^4 +
2*a*c*d^2*e^6 + a^2*e^8) - 75/16*c^4*d^3*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^6 + a*sqrt(c)*e^8) + 1/24*(c*x
^2 + a)^(7/2)*c^2*d^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) - 1/24*(c*x^2 + a)^(5/2)*c^3*d^2/(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) + 15/4*c
^3*d^2*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) +
5/4*(c*x^2 + a)^(3/2)*c^3*d^2/(c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7) - 35/24*(c*x^2 + a)^(3/2)*c^3*d*x/(c^2*d
^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) - 35/16*sqrt(c*x^2 + a)*a*c^3*d*x/(c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6) -
45/16*c^(5/2)*d*arcsinh(c*x/sqrt(a*c))*e^(-6) - 35/16*a*c^3*d*arcsinh(c*x/sqrt(a*c))/(c^(3/2)*d^2*e^4 + a*sqrt
(c)*e^6) - 1/12*(c*x^2 + a)^(7/2)*c*d/(c^2*d^4*x^3*e^2 + 3*c^2*d^5*x^2*e + c^2*d^7*e^(-1) + 3*c^2*d^6*x + 2*a*
c*d^2*x^3*e^4 + 6*a*c*d^3*x^2*e^3 + 6*a*c*d^4*x*e^2 + 2*a*c*d^5*e + a^2*x^3*e^6 + 3*a^2*d*x^2*e^5 + 3*a^2*d^2*
x*e^4 + a^2*d^3*e^3) + 19/24*(c*x^2 + a)^(5/2)*c^2*d/(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) + 15/4*sqrt(c*x^2 + a)*c^3*d^2/(c*d^2*e^5 + a*e^7) - 15/16*sqrt(c*x^2 + a)*c^3*d*
x/(c*d^2*e^4 + a*e^6) + 15/8*sqrt(c*d^2*e^(-2) + a)*c^2*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c
)*abs(x*e + d)))*e^(-5) - 3/8*(c*x^2 + a)^(7/2)*c/(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) + 3/8*(c*x^2 + a)^(5/2)*c^2
/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + 15/8*sqrt(c*x^2 + a)*c^2*e^(-5) - 1/4*(c*x^2 + a)^(7/2)/(c*d^2*x^4*e^
3 + 4*c*d^3*x^3*e^2 + 6*c*d^4*x^2*e + c*d^6*e^(-1) + 4*c*d^5*x + a*x^4*e^5 + 4*a*d*x^3*e^4 + 6*a*d^2*x^2*e^3 +
4*a*d^3*x*e^2 + a*d^4*e) + 5/8*(c*x^2 + a)^(3/2)*c^2/(c*d^2*e^3 + a*e^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 907 vs. \(2 (253) = 506\).
time = 23.01, size = 3695, normalized size = 12.87 \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)^5,x, algorithm="fricas")

[Out]

[1/48*(120*(4*c^4*d^8*x*e + c^4*d^9 + a^2*c^2*d*x^4*e^8 + 4*a^2*c^2*d^2*x^3*e^7 + 2*(a*c^3*d^3*x^4 + 3*a^2*c^2
*d^3*x^2)*e^6 + 4*(2*a*c^3*d^4*x^3 + a^2*c^2*d^4*x)*e^5 + (c^4*d^5*x^4 + 12*a*c^3*d^5*x^2 + a^2*c^2*d^5)*e^4 +
4*(c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e^3 + 2*(3*c^4*d^7*x^2 + a*c^3*d^7)*e^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2
+ a)*sqrt(c)*x - a) + 15*(32*c^4*d^7*x*e + 8*c^4*d^8 + 3*a^2*c^2*x^4*e^8 + 12*a^2*c^2*d*x^3*e^7 + 6*(2*a*c^3*d
^2*x^4 + 3*a^2*c^2*d^2*x^2)*e^6 + 12*(4*a*c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^5 + (8*c^4*d^4*x^4 + 72*a*c^3*d^4*x^2
+ 3*a^2*c^2*d^4)*e^4 + 16*(2*c^4*d^5*x^3 + 3*a*c^3*d^5*x)*e^3 + 12*(4*c^4*d^6*x^2 + a*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*(420*c^4*d^7*x*e^2 + 120*c^4*d^8*e + 3*(8*a^2*c^2*x^4 -
9*a^3*c*x^2 - 2*a^4)*e^9 + 5*(47*a^2*c^2*d*x^3 - 4*a^3*c*d*x)*e^8 + (48*a*c^3*d^2*x^4 + 421*a^2*c^2*d^2*x^2 -
17*a^3*c*d^2)*e^7 + 5*(97*a*c^3*d^3*x^3 + 67*a^2*c^2*d^3*x)*e^6 + (24*c^4*d^4*x^4 + 968*a*c^3*d^4*x^2 + 89*a^
2*c^2*d^4)*e^5 + 25*(10*c^4*d^5*x^3 + 31*a*c^3*d^5*x)*e^4 + 20*(26*c^4*d^6*x^2 + 11*a*c^3*d^6)*e^3)*sqrt(c*x^2
+ a))/(4*c^2*d^7*x*e^7 + c^2*d^8*e^6 + a^2*x^4*e^14 + 4*a^2*d*x^3*e^13 + 2*(a*c*d^2*x^4 + 3*a^2*d^2*x^2)*e^12
+ 4*(2*a*c*d^3*x^3 + a^2*d^3*x)*e^11 + (c^2*d^4*x^4 + 12*a*c*d^4*x^2 + a^2*d^4)*e^10 + 4*(c^2*d^5*x^3 + 2*a*c
*d^5*x)*e^9 + 2*(3*c^2*d^6*x^2 + a*c*d^6)*e^8), 1/48*(240*(4*c^4*d^8*x*e + c^4*d^9 + a^2*c^2*d*x^4*e^8 + 4*a^2
*c^2*d^2*x^3*e^7 + 2*(a*c^3*d^3*x^4 + 3*a^2*c^2*d^3*x^2)*e^6 + 4*(2*a*c^3*d^4*x^3 + a^2*c^2*d^4*x)*e^5 + (c^4*
d^5*x^4 + 12*a*c^3*d^5*x^2 + a^2*c^2*d^5)*e^4 + 4*(c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e^3 + 2*(3*c^4*d^7*x^2 + a*c^3
*d^7)*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 15*(32*c^4*d^7*x*e + 8*c^4*d^8 + 3*a^2*c^2*x^4*e^8 +
12*a^2*c^2*d*x^3*e^7 + 6*(2*a*c^3*d^2*x^4 + 3*a^2*c^2*d^2*x^2)*e^6 + 12*(4*a*c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^5
+ (8*c^4*d^4*x^4 + 72*a*c^3*d^4*x^2 + 3*a^2*c^2*d^4)*e^4 + 16*(2*c^4*d^5*x^3 + 3*a*c^3*d^5*x)*e^3 + 12*(4*c^4*
d^6*x^2 + a*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*(420*c^4*d^7*x*e^2 +
120*c^4*d^8*e + 3*(8*a^2*c^2*x^4 - 9*a^3*c*x^2 - 2*a^4)*e^9 + 5*(47*a^2*c^2*d*x^3 - 4*a^3*c*d*x)*e^8 + (48*a*
c^3*d^2*x^4 + 421*a^2*c^2*d^2*x^2 - 17*a^3*c*d^2)*e^7 + 5*(97*a*c^3*d^3*x^3 + 67*a^2*c^2*d^3*x)*e^6 + (24*c^4*
d^4*x^4 + 968*a*c^3*d^4*x^2 + 89*a^2*c^2*d^4)*e^5 + 25*(10*c^4*d^5*x^3 + 31*a*c^3*d^5*x)*e^4 + 20*(26*c^4*d^6*
x^2 + 11*a*c^3*d^6)*e^3)*sqrt(c*x^2 + a))/(4*c^2*d^7*x*e^7 + c^2*d^8*e^6 + a^2*x^4*e^14 + 4*a^2*d*x^3*e^13 + 2
*(a*c*d^2*x^4 + 3*a^2*d^2*x^2)*e^12 + 4*(2*a*c*d^3*x^3 + a^2*d^3*x)*e^11 + (c^2*d^4*x^4 + 12*a*c*d^4*x^2 + a^2
*d^4)*e^10 + 4*(c^2*d^5*x^3 + 2*a*c*d^5*x)*e^9 + 2*(3*c^2*d^6*x^2 + a*c*d^6)*e^8), 1/24*(15*(32*c^4*d^7*x*e +
8*c^4*d^8 + 3*a^2*c^2*x^4*e^8 + 12*a^2*c^2*d*x^3*e^7 + 6*(2*a*c^3*d^2*x^4 + 3*a^2*c^2*d^2*x^2)*e^6 + 12*(4*a*c
^3*d^3*x^3 + a^2*c^2*d^3*x)*e^5 + (8*c^4*d^4*x^4 + 72*a*c^3*d^4*x^2 + 3*a^2*c^2*d^4)*e^4 + 16*(2*c^4*d^5*x^3 +
3*a*c^3*d^5*x)*e^3 + 12*(4*c^4*d^6*x^2 + a*c^3*d^6)*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)) + 60*(4*c^4*d^8*x*e + c^4*d^9 + a^2
*c^2*d*x^4*e^8 + 4*a^2*c^2*d^2*x^3*e^7 + 2*(a*c^3*d^3*x^4 + 3*a^2*c^2*d^3*x^2)*e^6 + 4*(2*a*c^3*d^4*x^3 + a^2*
c^2*d^4*x)*e^5 + (c^4*d^5*x^4 + 12*a*c^3*d^5*x^2 + a^2*c^2*d^5)*e^4 + 4*(c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e^3 + 2*
(3*c^4*d^7*x^2 + a*c^3*d^7)*e^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (420*c^4*d^7*x*e^2
+ 120*c^4*d^8*e + 3*(8*a^2*c^2*x^4 - 9*a^3*c*x^2 - 2*a^4)*e^9 + 5*(47*a^2*c^2*d*x^3 - 4*a^3*c*d*x)*e^8 + (48*a
*c^3*d^2*x^4 + 421*a^2*c^2*d^2*x^2 - 17*a^3*c*d^2)*e^7 + 5*(97*a*c^3*d^3*x^3 + 67*a^2*c^2*d^3*x)*e^6 + (24*c^4
*d^4*x^4 + 968*a*c^3*d^4*x^2 + 89*a^2*c^2*d^4)*e^5 + 25*(10*c^4*d^5*x^3 + 31*a*c^3*d^5*x)*e^4 + 20*(26*c^4*d^6
*x^2 + 11*a*c^3*d^6)*e^3)*sqrt(c*x^2 + a))/(4*c^2*d^7*x*e^7 + c^2*d^8*e^6 + a^2*x^4*e^14 + 4*a^2*d*x^3*e^13 +
2*(a*c*d^2*x^4 + 3*a^2*d^2*x^2)*e^12 + 4*(2*a*c*d^3*x^3 + a^2*d^3*x)*e^11 + (c^2*d^4*x^4 + 12*a*c*d^4*x^2 + a^
2*d^4)*e^10 + 4*(c^2*d^5*x^3 + 2*a*c*d^5*x)*e^9 + 2*(3*c^2*d^6*x^2 + a*c*d^6)*e^8), 1/24*(15*(32*c^4*d^7*x*e +
8*c^4*d^8 + 3*a^2*c^2*x^4*e^8 + 12*a^2*c^2*d*x^3*e^7 + 6*(2*a*c^3*d^2*x^4 + 3*a^2*c^2*d^2*x^2)*e^6 + 12*(4*a*
c^3*d^3*x^3 + a^2*c^2*d^3*x)*e^5 + (8*c^4*d^4*x^4 + 72*a*c^3*d^4*x^2 + 3*a^2*c^2*d^4)*e^4 + 16*(2*c^4*d^5*x^3
+ 3*a*c^3*d^5*x)*e^3 + 12*(4*c^4*d^6*x^2 + a*c^3*d^6)*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)) + 120*(4*c^4*d^8*x*e + c^4*d^9 + a
^2*c^2*d*x^4*e^8 + 4*a^2*c^2*d^2*x^3*e^7 + 2*(a*c^3*d^3*x^4 + 3*a^2*c^2*d^3*x^2)*e^6 + 4*(2*a*c^3*d^4*x^3 + a^
2*c^2*d^4*x)*e^5 + (c^4*d^5*x^4 + 12*a*c^3*d^5*...

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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 )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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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 )}^5} \,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)^5,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^5, x)

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