3.543 \(\int \frac{(a+c x^2)^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=195 \[ -\frac{3 a^2 c^3 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{3 a c^2 d \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \]

[Out]

(-3*a*c^2*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (c*d*(a*e - c*d*x)*(a + c*x^2)^
(3/2))/(4*(c*d^2 + a*e^2)^2*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(5*(c*d^2 + a*e^2)*(d + e*x)^5) - (3*a^2*c^3*
d*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)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0950982, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {731, 721, 725, 206} \[ -\frac{3 a^2 c^3 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{3 a c^2 d \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(-3*a*c^2*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (c*d*(a*e - c*d*x)*(a + c*x^2)^
(3/2))/(4*(c*d^2 + a*e^2)^2*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(5*(c*d^2 + a*e^2)*(d + e*x)^5) - (3*a^2*c^3*
d*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)^(7/2))

Rule 731

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

Rule 721

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 725

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx &=-\frac{e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac{(c d) \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{c d^2+a e^2}\\ &=-\frac{c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac{\left (3 a c^2 d\right ) \int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx}{4 \left (c d^2+a e^2\right )^2}\\ &=-\frac{3 a c^2 d (a e-c d x) \sqrt{a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac{\left (3 a^2 c^3 d\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac{3 a c^2 d (a e-c d x) \sqrt{a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac{\left (3 a^2 c^3 d\right ) \operatorname{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 )^3}\\ &=-\frac{3 a c^2 d (a e-c d x) \sqrt{a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac{3 a^2 c^3 d \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 )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.374942, size = 272, normalized size = 1.39 \[ \frac{1}{40} \left (\frac{\sqrt{a+c x^2} \left (-a^2 c^2 e \left (77 d^2 e^2 x^2+45 d^3 e x+33 d^4+25 d e^3 x^3+8 e^4 x^4\right )-2 a^3 c e^3 \left (13 d^2+5 d e x+8 e^2 x^2\right )-8 a^4 e^5+a c^3 d^2 x \left (29 d^2 e x+25 d^3+45 d e^2 x^2+9 e^3 x^3\right )+2 c^4 d^4 x^3 (5 d+e x)\right )}{(d+e x)^5 \left (a e^2+c d^2\right )^3}-\frac{15 a^2 c^3 d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{15 a^2 c^3 d \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

((Sqrt[a + c*x^2]*(-8*a^4*e^5 + 2*c^4*d^4*x^3*(5*d + e*x) - 2*a^3*c*e^3*(13*d^2 + 5*d*e*x + 8*e^2*x^2) + a*c^3
*d^2*x*(25*d^3 + 29*d^2*e*x + 45*d*e^2*x^2 + 9*e^3*x^3) - a^2*c^2*e*(33*d^4 + 45*d^3*e*x + 77*d^2*e^2*x^2 + 25
*d*e^3*x^3 + 8*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d + e*x)^5) + (15*a^2*c^3*d*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2)
- (15*a^2*c^3*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/40

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Maple [B]  time = 0.217, size = 3622, normalized size = 18.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^6,x)

[Out]

-1/8*c^5*d^4/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x+1/8*c^4*d^4/(a*e^2+c*d^2)
^5/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-3/16*c^(9/2)*d^4/(a*e^2+c*d^2)^5*a^2*ln((-c*d
/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+9/16*c^(7/2)*d^2/(a*e^2+c*d^2)^4*
a^2*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-3/8*c^3*d^2/(a*e^2+c*
d^2)^4/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-1/5/e^4/(a*e^2+c*d^2)/(d/e+x)^5*(c*(d/e+x
)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+1/8/e*c^5*d^5/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2
+c*d^2)/e^2)^(3/2)+3/8/e^3*c^6*d^7/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-3/8/e
^4*c^(9/2)*d^4/(a*e^2+c*d^2)^3*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
1/2))+3/4/e^4*c^(11/2)*d^6/(a*e^2+c*d^2)^4*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c
*d^2)/e^2)^(1/2))-3/8/e^4*c^(13/2)*d^8/(a*e^2+c*d^2)^5*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e
+x)+(a*e^2+c*d^2)/e^2)^(1/2))+3/8*c^4*d^2/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2
)*x-3/4/e^3*c^5*d^5/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-1/4/e*c^4*d^3/(a*e^2
+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+1/8/e*c^3*d/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d
/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+3/8/e^3*c^4*d^3/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)
/e^2)^(1/2)-3/8/e*c^3*d/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((
a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2-3/4/e^3*c^4*d^3/(a
*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c
*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a-3/8/e*c^5*d^5/(a*e^2+c*d^2)^5/((a*e^2+c*d^2)/e
^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2-3/4/e^3*c^6*d^7/(a*e^2+c*d^2)^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d
^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/
e+x))*a+3/4/e*c^4*d^3/(a*e^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*
e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2+3/2/e^3*c^5*d^5/(a*e
^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(
d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a+3/4/e^5*c^6*d^7/(a*e^2+c*d^2)^4/((a*e^2+c*d^2)/e
^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(1/2))/(d/e+x))-1/4/e^3*c*d/(a*e^2+c*d^2)^2/(d/e+x)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^
2)/e^2)^(5/2)-1/8/e*c^2*d/(a*e^2+c*d^2)^3/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-3/16
*c^5*d^4/(a*e^2+c*d^2)^5*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+9/16*c^4*d^2/(a*e^2+c*d^2)^
4*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+3/8/e*c^3*d/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(
d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a-1/4/e^2*c^2*d^2/(a*e^2+c*d^2)^3/(d/e+x)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^
2+c*d^2)/e^2)^(5/2)+3/8/e^2*c^5*d^4/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/
4/e*c^4*d^3/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a-1/8/e*c^3*d^3/(a*e^2+c*d^2
)^4/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-9/16/e^2*c^(11/2)*d^6/(a*e^2+c*d^2)^5*ln((
-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a-9/16/e^2*c^(7/2)*d^2/(a*e^2
+c*d^2)^3*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a+9/8/e^2*c^(9/
2)*d^4/(a*e^2+c*d^2)^4*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a-
3/16/e^2*c^6*d^6/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+3/8/e*c^5*d^5/(a*e^2+
c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a-3/8/e^5*c^5*d^5/(a*e^2+c*d^2)^3/((a*e^2+c*d^2
)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+
(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))-3/16/e^2*c^4*d^2/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)
/e^2)^(1/2)*x-3/8/e^5*c^7*d^9/(a*e^2+c*d^2)^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x
)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 74.4061, size = 3316, normalized size = 17.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

[1/80*(15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e^3*x^3 + 10*a^2*c^3*d^4*e^2*x^2 + 5*a^2
*c^3*d^5*e*x + a^2*c^3*d^6)*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*(33*a^2*c^3*d^6*e +
59*a^3*c^2*d^4*e^3 + 34*a^4*c*d^2*e^5 + 8*a^5*e^7 - (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3
*c^2*e^7)*x^4 - 5*(2*c^5*d^7 + 11*a*c^4*d^5*e^2 + 4*a^2*c^3*d^3*e^4 - 5*a^3*c^2*d*e^6)*x^3 - (29*a*c^4*d^6*e -
48*a^2*c^3*d^4*e^3 - 93*a^3*c^2*d^2*e^5 - 16*a^4*c*e^7)*x^2 - 5*(5*a*c^4*d^7 - 4*a^2*c^3*d^5*e^2 - 11*a^3*c^2
*d^3*e^4 - 2*a^4*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^13 + 4*a*c^3*d^11*e^2 + 6*a^2*c^2*d^9*e^4 + 4*a^3*c*d^7*e
^6 + a^4*d^5*e^8 + (c^4*d^8*e^5 + 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13)*x^5 + 5*(
c^4*d^9*e^4 + 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 + 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^4 + 10*(c^4*d^10*e^3 + 4*
a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 + 4*a^3*c*d^4*e^9 + a^4*d^2*e^11)*x^3 + 10*(c^4*d^11*e^2 + 4*a*c^3*d^9*e^4 +
6*a^2*c^2*d^7*e^6 + 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e + 4*a*c^3*d^10*e^3 + 6*a^2*c^2*d^8*e^
5 + 4*a^3*c*d^6*e^7 + a^4*d^4*e^9)*x), -1/40*(15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e
^3*x^3 + 10*a^2*c^3*d^4*e^2*x^2 + 5*a^2*c^3*d^5*e*x + a^2*c^3*d^6)*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)) + (33*a^2*c^3*d^6*e + 59*a^
3*c^2*d^4*e^3 + 34*a^4*c*d^2*e^5 + 8*a^5*e^7 - (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3*c^2*e
^7)*x^4 - 5*(2*c^5*d^7 + 11*a*c^4*d^5*e^2 + 4*a^2*c^3*d^3*e^4 - 5*a^3*c^2*d*e^6)*x^3 - (29*a*c^4*d^6*e - 48*a^
2*c^3*d^4*e^3 - 93*a^3*c^2*d^2*e^5 - 16*a^4*c*e^7)*x^2 - 5*(5*a*c^4*d^7 - 4*a^2*c^3*d^5*e^2 - 11*a^3*c^2*d^3*e
^4 - 2*a^4*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^13 + 4*a*c^3*d^11*e^2 + 6*a^2*c^2*d^9*e^4 + 4*a^3*c*d^7*e^6 + a
^4*d^5*e^8 + (c^4*d^8*e^5 + 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13)*x^5 + 5*(c^4*d^
9*e^4 + 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 + 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^4 + 10*(c^4*d^10*e^3 + 4*a*c^3*
d^8*e^5 + 6*a^2*c^2*d^6*e^7 + 4*a^3*c*d^4*e^9 + a^4*d^2*e^11)*x^3 + 10*(c^4*d^11*e^2 + 4*a*c^3*d^9*e^4 + 6*a^2
*c^2*d^7*e^6 + 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e + 4*a*c^3*d^10*e^3 + 6*a^2*c^2*d^8*e^5 + 4*
a^3*c*d^6*e^7 + a^4*d^4*e^9)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**6, x)

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Giac [B]  time = 1.45803, size = 1682, normalized size = 8.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

-3/4*a^2*c^3*d*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*
d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + 1/20*(80*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)
*d^8*e + 32*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*d^9 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*d^7*e^2 + 40*(sqr
t(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*d^6*e^3 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d^8*e - 16*(sqrt(
c)*x - sqrt(c*x^2 + a))^5*a*c^6*d^7*e^2 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*d^6*e^3 + 240*(sqrt(c
)*x - sqrt(c*x^2 + a))^7*a*c^5*d^5*e^4 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*d^7*e^2 + 120*(sqrt(c)*x -
sqrt(c*x^2 + a))^8*a*c^(9/2)*d^4*e^5 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*d^6*e^3 - 788*(sqrt(c
)*x - sqrt(c*x^2 + a))^5*a^2*c^5*d^5*e^4 - 530*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*d^4*e^5 - 40*(sqrt(
c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*d^6*e^3 - 230*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*d^3*e^6 + 400*(sq
rt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*d^5*e^4 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d^2*e^7 + 1170*(
sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*d^4*e^5 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*d*e^8 + 910*(s
qrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*d^3*e^6 + 20*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c^5*d^5*e^4 + 570*(sqrt(c
)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*d^2*e^7 - 230*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(9/2)*d^4*e^5 + 150*(
sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*e^8 - 770*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*d^3*e^6 + 40*(sqrt(
c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e^9 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*d^2*e^7 - 2*a^5*c^
(9/2)*d^4*e^5 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*d*e^8 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^4*d
^3*e^6 + 350*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*d^2*e^7 + 170*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3
*d*e^8 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(5/2)*e^9 - 9*a^6*c^(7/2)*d^2*e^7 - 65*(sqrt(c)*x - sqrt(c*x
^2 + a))*a^6*c^3*d*e^8 + 8*a^7*c^(5/2)*e^9)/((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)*((sq
rt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^5)