### 3.568 $$\int \frac{1}{(d+e x)^4 \sqrt{a+c x^2}} \, dx$$

Optimal. Leaf size=198 $-\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac{c e \sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}$

[Out]

-(e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (5*c*d*e*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^2*(d + e*x
)^2) - (c*e*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^3*(d + e*x)) - (c^2*d*(2*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*(c*d^2 + a*e^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.16382, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.263, Rules used = {745, 835, 807, 725, 206} $-\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac{c e \sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]

[Out]

-(e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (5*c*d*e*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^2*(d + e*x
)^2) - (c*e*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^3*(d + e*x)) - (c^2*d*(2*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*(c*d^2 + a*e^2)^(7/2))

Rule 745

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/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 835

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

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 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{1}{(d+e x)^4 \sqrt{a+c x^2}} \, dx &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{c \int \frac{-3 d+2 e x}{(d+e x)^3 \sqrt{a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac{c \int \frac{2 \left (3 c d^2-2 a e^2\right )-5 c d e x}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{6 \left (c d^2+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c e \left (11 c d^2-4 a e^2\right ) \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac{\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^3}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c e \left (11 c d^2-4 a e^2\right ) \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{\left (c^2 d \left (2 c d^2-3 a e^2\right )\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 )}{2 \left (c d^2+a e^2\right )^3}\\ &=-\frac{e \sqrt{a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c e \left (11 c d^2-4 a e^2\right ) \sqrt{a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac{c^2 d \left (2 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 \left (c d^2+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.178611, size = 209, normalized size = 1.06 $\frac{-3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log (d+e x)-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (5 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (11 c d^2-4 a e^2\right )+2 \left (a e^2+c d^2\right )^2\right )}{6 (d+e x)^3 \left (a e^2+c d^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]

[Out]

(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(2*(c*d^2 + a*e^2)^2 + 5*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2
- 4*a*e^2)*(d + e*x)^2)) + 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[d + e*x] - 3*c^2*d*(2*c*d^2 - 3*a*e^2)*
(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^3)

________________________________________________________________________________________

Maple [B]  time = 0.196, size = 573, normalized size = 2.9 \begin{align*} -{\frac{1}{3\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{5\,cd}{6\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{5\,{c}^{2}{d}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{5\,{c}^{3}{d}^{3}}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{3\,{c}^{2}d}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{2\,c}{3\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(c*x^2+a)^(1/2),x)

[Out]

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

________________________________________________________________________________________

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(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 8.50196, size = 2290, normalized size = 11.57 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(2*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*
e^4)*x^2 + 3*(2*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
*(18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*
e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 +
6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^
3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e^6 + 4*a^3*c*d^3*e^8 + a^4*d*e
^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x), -1/6*(3*(2
*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 + 3
*(2*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)) + (18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a
^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e
^6)*x)*sqrt(c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4
*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^
7*e^4 + 6*a^2*c^2*d^5*e^6 + 4*a^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^
6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + c x^{2}} \left (d + e x\right )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**4/(c*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**2)*(d + e*x)**4), x)

________________________________________________________________________________________

Giac [B]  time = 1.37986, size = 780, normalized size = 3.94 \begin{align*} \frac{1}{3} \, c^{\frac{3}{2}}{\left (\frac{3 \,{\left (2 \, c^{\frac{3}{2}} d^{3} - 3 \, a \sqrt{c} 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 )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} c^{2} d^{4} e + 44 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c^{\frac{5}{2}} d^{5} + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} c^{\frac{3}{2}} d^{3} e^{2} - 102 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a c^{2} d^{4} e - 82 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a c^{\frac{3}{2}} d^{3} e^{2} - 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a c d^{2} e^{3} - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a \sqrt{c} d e^{4} + 60 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} c^{\frac{3}{2}} d^{3} e^{2} + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{2} \sqrt{c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} - 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{3} \sqrt{c} d e^{4} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{3} e^{5} + 4 \, a^{4} e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\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}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/3*c^(3/2)*(3*(2*c^(3/2)*d^3 - 3*a*sqrt(c)*d*e^2)*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)) - (30*(sqrt(c)
*x - sqrt(c*x^2 + a))^4*c^2*d^4*e + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^(5/2)*d^5 + 6*(sqrt(c)*x - sqrt(c*x^2
+ a))^5*c^(3/2)*d^3*e^2 - 102*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^2*d^4*e - 82*(sqrt(c)*x - sqrt(c*x^2 + a))^
3*a*c^(3/2)*d^3*e^2 - 45*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*sqr
t(c)*d*e^4 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c*d
^2*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*sqrt(c)*d*e^4 - 11*a^3*c*d^2*e^3 - 15*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^3*sqrt(c)*d*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*e^5 + 4*a^4*e^5)/((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)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d -
a*e)^3))