\(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^5} \, dx\) [542]

Optimal result

Integrand size = 19, antiderivative size = 153 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {3 a c (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {3 a^2 c^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {735, 739, 212} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {3 a^2 c^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac {3 a c \sqrt {a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}+\frac {(3 a c) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{4 \left (c d^2+a e^2\right )} \\ & = -\frac {3 a c (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}+\frac {\left (3 a^2 c^2\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2} \\ & = -\frac {3 a c (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {\left (3 a^2 c^2\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 \left (c d^2+a e^2\right )^2} \\ & = -\frac {3 a c (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {3 a^2 c^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{5/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(723\) vs. \(2(153)=306\).

Time = 7.52 (sec) , antiderivative size = 723, normalized size of antiderivative = 4.73 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {1}{8} \left (\frac {16 c^5 d^4 x^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right ) \left (\sqrt {c} x-\sqrt {a+c x^2}\right )-2 a^5 e^7 \left (-4 \sqrt {c} x+\sqrt {a+c x^2}\right )+a^4 c e^5 \left (4 \sqrt {c} x \left (5 d^2+4 d e x+11 e^2 x^2\right )-\sqrt {a+c x^2} \left (5 d^2+4 d e x+21 e^2 x^2\right )\right )-8 a c^4 d^2 x \left (\sqrt {a+c x^2} \left (d^5+4 d^4 e x+11 d^3 e^2 x^2+24 d^2 e^3 x^3+24 d e^4 x^4+16 e^5 x^5\right )-\sqrt {c} x \left (2 d^5+8 d^4 e x+17 d^3 e^2 x^2+28 d^2 e^3 x^3+24 d e^4 x^4+16 e^5 x^5\right )\right )+a^3 \left (-c^2 e^4 x \sqrt {a+c x^2} \left (-5 d^3+36 d^2 e x+27 d e^2 x^2+56 e^3 x^3\right )+c^{5/2} e^2 \left (5 d^5+20 d^4 e x+10 d^3 e^2 x^2+64 d^2 e^3 x^3+33 d e^4 x^4+76 e^5 x^5\right )\right )+2 a^2 \left (-c^3 e^2 x \sqrt {a+c x^2} \left (10 d^5+40 d^4 e x+39 d^3 e^2 x^2+44 d^2 e^3 x^3+6 d e^4 x^4+20 e^5 x^5\right )+c^{7/2} \left (d^7+4 d^6 e x+26 d^5 e^2 x^2+84 d^4 e^3 x^3+87 d^3 e^4 x^4+76 d^2 e^5 x^5+6 d e^6 x^6+20 e^7 x^7\right )\right )}{e^4 \left (c d^2+a e^2\right )^2 (d+e x)^4 \left (a^2+8 a c x^2+8 c^2 x^4-4 a \sqrt {c} x \sqrt {a+c x^2}-8 c^{3/2} x^3 \sqrt {a+c x^2}\right )}-\frac {6 a^2 c^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{5/2}}\right ) \]

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

((16*c^5*d^4*x^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)*(Sqrt[c]*x - Sqrt[a + c*x^2]) - 2*a^5*e^7*(-4*Sqr
t[c]*x + Sqrt[a + c*x^2]) + a^4*c*e^5*(4*Sqrt[c]*x*(5*d^2 + 4*d*e*x + 11*e^2*x^2) - Sqrt[a + c*x^2]*(5*d^2 + 4
*d*e*x + 21*e^2*x^2)) - 8*a*c^4*d^2*x*(Sqrt[a + c*x^2]*(d^5 + 4*d^4*e*x + 11*d^3*e^2*x^2 + 24*d^2*e^3*x^3 + 24
*d*e^4*x^4 + 16*e^5*x^5) - Sqrt[c]*x*(2*d^5 + 8*d^4*e*x + 17*d^3*e^2*x^2 + 28*d^2*e^3*x^3 + 24*d*e^4*x^4 + 16*
e^5*x^5)) + a^3*(-(c^2*e^4*x*Sqrt[a + c*x^2]*(-5*d^3 + 36*d^2*e*x + 27*d*e^2*x^2 + 56*e^3*x^3)) + c^(5/2)*e^2*
(5*d^5 + 20*d^4*e*x + 10*d^3*e^2*x^2 + 64*d^2*e^3*x^3 + 33*d*e^4*x^4 + 76*e^5*x^5)) + 2*a^2*(-(c^3*e^2*x*Sqrt[
a + c*x^2]*(10*d^5 + 40*d^4*e*x + 39*d^3*e^2*x^2 + 44*d^2*e^3*x^3 + 6*d*e^4*x^4 + 20*e^5*x^5)) + c^(7/2)*(d^7
+ 4*d^6*e*x + 26*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 87*d^3*e^4*x^4 + 76*d^2*e^5*x^5 + 6*d*e^6*x^6 + 20*e^7*x^7)))/
(e^4*(c*d^2 + a*e^2)^2*(d + e*x)^4*(a^2 + 8*a*c*x^2 + 8*c^2*x^4 - 4*a*Sqrt[c]*x*Sqrt[a + c*x^2] - 8*c^(3/2)*x^
3*Sqrt[a + c*x^2])) - (6*a^2*c^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)^(5/2))/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4019\) vs. \(2(137)=274\).

Time = 2.19 (sec) , antiderivative size = 4020, normalized size of antiderivative = 26.27

method	result	size
default	\(\text {Expression too large to display}\)	\(4020\)

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

1/e^5*(-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)^(5/2)+3/4*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)^(5/2)+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)^(5/2)-1/2*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)^(5/2)-3*c*d*e/(a*e^2
+c*d^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*c/(a*e^2+c*d^2)*e^
2*(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)))))+3/2*c/(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)))))+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)^(5/2)-3*c*d*e/(a*e^2+c*d^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*c/(a*e^2+c*d^2)*e^2*(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))))))+1/4*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)^(5/2)-1/2*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)^(5/2)-3*c*d*e/(a*e^2+c*d^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*c/(a*e^2+c*d^2)*e^2*(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)*l
n((-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)))))+3/2*c/(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))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (138) = 276\).

Time = 1.26 (sec) , antiderivative size = 1123, normalized size of antiderivative = 7.34 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\left [\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 4 \, a^{2} c^{2} d e^{3} x^{3} + 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 4 \, a^{2} c^{2} d^{3} e x + a^{2} c^{2} d^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (5 \, a^{2} c^{2} d^{4} e + 7 \, a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5} - {\left (2 \, c^{4} d^{5} + 7 \, a c^{3} d^{3} e^{2} + 5 \, a^{2} c^{2} d e^{4}\right )} x^{3} - {\left (4 \, a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} - 5 \, a^{3} c e^{5}\right )} x^{2} - {\left (5 \, a c^{3} d^{5} + a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{16 \, {\left (c^{3} d^{10} + 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} + a^{3} d^{4} e^{6} + {\left (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}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} + 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} + a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} + 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} + a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e + 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} + a^{3} d^{3} e^{7}\right )} x\right )}}, -\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 4 \, a^{2} c^{2} d e^{3} x^{3} + 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 4 \, a^{2} c^{2} d^{3} e x + a^{2} c^{2} d^{4}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (5 \, a^{2} c^{2} d^{4} e + 7 \, a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5} - {\left (2 \, c^{4} d^{5} + 7 \, a c^{3} d^{3} e^{2} + 5 \, a^{2} c^{2} d e^{4}\right )} x^{3} - {\left (4 \, a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} - 5 \, a^{3} c e^{5}\right )} x^{2} - {\left (5 \, a c^{3} d^{5} + a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{8 \, {\left (c^{3} d^{10} + 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} + a^{3} d^{4} e^{6} + {\left (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}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} + 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} + a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} + 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} + a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e + 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} + a^{3} d^{3} e^{7}\right )} x\right )}}\right ] \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

[1/16*(3*(a^2*c^2*e^4*x^4 + 4*a^2*c^2*d*e^3*x^3 + 6*a^2*c^2*d^2*e^2*x^2 + 4*a^2*c^2*d^3*e*x + a^2*c^2*d^4)*sqr
t(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*(5*a^2*c^2*d^4*e + 7*a^3*c*d^2*e^3 + 2*a^4*e^5 - (
2*c^4*d^5 + 7*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4)*x^3 - (4*a*c^3*d^4*e - a^2*c^2*d^2*e^3 - 5*a^3*c*e^5)*x^2 - (5*
a*c^3*d^5 + a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^10 + 3*a*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4
+ a^3*d^4*e^6 + (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)*x^4 + 4*(c^3*d^7*e^3 + 3*a*c^2*d
^5*e^5 + 3*a^2*c*d^3*e^7 + a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 + a^3*d^2*e^8)*
x^2 + 4*(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 + a^3*d^3*e^7)*x), -1/8*(3*(a^2*c^2*e^4*x^4 + 4*a^2*c^2
*d*e^3*x^3 + 6*a^2*c^2*d^2*e^2*x^2 + 4*a^2*c^2*d^3*e*x + a^2*c^2*d^4)*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)) + (5*a^2*c^2*d^4*e + 7*a
^3*c*d^2*e^3 + 2*a^4*e^5 - (2*c^4*d^5 + 7*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4)*x^3 - (4*a*c^3*d^4*e - a^2*c^2*d^2*
e^3 - 5*a^3*c*e^5)*x^2 - (5*a*c^3*d^5 + a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^10 + 3*a*c
^2*d^8*e^2 + 3*a^2*c*d^6*e^4 + a^3*d^4*e^6 + (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)*x^4
+ 4*(c^3*d^7*e^3 + 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 + a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a
^2*c*d^4*e^6 + a^3*d^2*e^8)*x^2 + 4*(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 + a^3*d^3*e^7)*x)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{5}}\, dx \]

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**5,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**5, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(3/2)/(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]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{5}} \,d x } \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^5} \,d x \]

[In]

int((a + c*x^2)^(3/2)/(d + e*x)^5,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^5, x)