### 3.541 $$\int \frac{(a+c x^2)^{3/2}}{(d+e x)^4} \, dx$$

Optimal. Leaf size=200 $\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \left (a e^2+c d^2\right )^{3/2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{c \sqrt{a+c x^2} \left (e x \left (2 a e^2+3 c d^2\right )+d \left (a e^2+2 c d^2\right )\right )}{2 e^3 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3}$

[Out]

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

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Rubi [A]  time = 0.176935, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.316, Rules used = {733, 811, 844, 217, 206, 725} $\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \left (a e^2+c d^2\right )^{3/2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{c \sqrt{a+c x^2} \left (e x \left (2 a e^2+3 c d^2\right )+d \left (a e^2+2 c d^2\right )\right )}{2 e^3 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

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

Rule 733

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 811

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*((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))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), 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 844

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]

Rule 217

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 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])

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]

Rubi steps

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

Mathematica [A]  time = 0.282589, size = 242, normalized size = 1.21 $\frac{-\frac{e \sqrt{a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+9 d e x+8 e^2 x^2\right )+c^2 d^2 \left (6 d^2+15 d e x+11 e^2 x^2\right )\right )}{(d+e x)^3 \left (a e^2+c d^2\right )}+\frac{3 c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{3 c^2 d \left (3 a e^2+2 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}+6 c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 e^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

(-((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 9*d*e*x + 8*e^2*x^2) + c^2*d^2*(6*d^2 + 15*d*e*x + 11*e^2*
x^2)))/((c*d^2 + a*e^2)*(d + e*x)^3)) - (3*c^2*d*(2*c*d^2 + 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + 6*c
^(3/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c^2*d*(2*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]
*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(6*e^4)

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Maple [B]  time = 0.216, size = 2490, normalized size = 12.5 \begin{align*} \text{result 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)^4,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)^(5/2)+2/3/(a*e^2+c*d^2)^2*c^2
*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*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)^(5/2)+1/(a*e^2+c*d^2)^2*c^(3/2)*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))+1/(a*e^2+c*d^2)^2*c^2*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1
/2)*x-3/4/e^2*c^(7/2)*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))*a+9/4/e^2*c^(5/2)*d^2/(a*e^2+c*d^2)^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))*a-1/2/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))*
a^2+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))*a^2+3/e^3*c^3*d^3/(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))*a-1/e^3*c^4*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))*a+3/4/e^2*c^3*d^2/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*
x-3/2/e*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a-1/6/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)^(5/2)-1/4/e^2*c^4*d^4/(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+1/2/e*c^3*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)*a-1/2/e^5*c^5*d^7/(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))-1/4*c^(5/2
)*d^2/(a*e^2+c*d^2)^3*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))
-1/6*c^3*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)^(3/2)*x+1/6*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)^(5/2)+1/6/e*c^3*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)^(3/2)+1/2/e^3*c^4*d^5/(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)-1/2/e*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-3/2/e^3*c^3*d
^3/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+3/2/e^4*c^(7/2)*d^4/(a*e^2+c*d^2)^2*l
n((-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))-1/2/e^4*c^(9/2)*d^6/(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))+3/2/e^5*c^4*d^5
/(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))-1/4*c^3*d^2/(a*e^2+c*d^2)^3*a*(c*(d/e+x)^2-2*
c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 54.1702, size = 5117, normalized size = 25.58 \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)^4,x, algorithm="fricas")

[Out]

[1/12*(6*(c^3*d^7 + 2*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (c^3*d^4*e^3 + 2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(c^3
*d^5*e^2 + 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(c^3*d^6*e + 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(c)*log
(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 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*(6*c^3*d^6*e + 11*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 + 19*a*c^2*d^2*e^5 + 8*a^2*c*e^7)*x^2 + 3*(5*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 + 3*a^2*c*d*e^6)*x)*sqrt(
c*x^2 + a))/(c^2*d^7*e^4 + 2*a*c*d^5*e^6 + a^2*d^3*e^8 + (c^2*d^4*e^7 + 2*a*c*d^2*e^9 + a^2*e^11)*x^3 + 3*(c^2
*d^5*e^6 + 2*a*c*d^3*e^8 + a^2*d*e^10)*x^2 + 3*(c^2*d^6*e^5 + 2*a*c*d^4*e^7 + a^2*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)) + 3*(c^3*d^7 + 2*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (c^3*d^4*e^3 +
2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(c^3*d^5*e^2 + 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(c^3*d^6*e + 2*a*c^
2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - (6*c^3*d^6*e + 11*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 + 19*a*c^2*d^2*e^5 + 8*a^2*c*e^7)*x^2 + 3*(5*c^3*d^5
*e^2 + 8*a*c^2*d^3*e^4 + 3*a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^2*d^7*e^4 + 2*a*c*d^5*e^6 + a^2*d^3*e^8 + (c^2*
d^4*e^7 + 2*a*c*d^2*e^9 + a^2*e^11)*x^3 + 3*(c^2*d^5*e^6 + 2*a*c*d^3*e^8 + a^2*d*e^10)*x^2 + 3*(c^2*d^6*e^5 +
2*a*c*d^4*e^7 + a^2*d^2*e^9)*x), -1/12*(12*(c^3*d^7 + 2*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (c^3*d^4*e^3 + 2*a*c^2
*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(c^3*d^5*e^2 + 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 3*(c^3*d^6*e + 2*a*c^2*d^4*e
^3 + a^2*c*d^2*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 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)*s
qrt(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*(6*c^3*d^6*e + 11*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 + 19*a*c^2*d^2*e^5 + 8*a^2*c*e^7)*x^2 + 3*(5*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 + 3
*a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^2*d^7*e^4 + 2*a*c*d^5*e^6 + a^2*d^3*e^8 + (c^2*d^4*e^7 + 2*a*c*d^2*e^9 +
a^2*e^11)*x^3 + 3*(c^2*d^5*e^6 + 2*a*c*d^3*e^8 + a^2*d*e^10)*x^2 + 3*(c^2*d^6*e^5 + 2*a*c*d^4*e^7 + a^2*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)) - 6*(c^3*d^7 + 2*a*c^2*d^5*e^2 + a^2*c*d^3
*e^4 + (c^3*d^4*e^3 + 2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^3 + 3*(c^3*d^5*e^2 + 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 +
3*(c^3*d^6*e + 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (6*c^3*d^6*e
+ 11*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 + 19*a*c^2*d^2*e^5 + 8*a^2*c*e^7)*x^2 + 3*
(5*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 + 3*a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^2*d^7*e^4 + 2*a*c*d^5*e^6 + a^2*d^3*e
^8 + (c^2*d^4*e^7 + 2*a*c*d^2*e^9 + a^2*e^11)*x^3 + 3*(c^2*d^5*e^6 + 2*a*c*d^3*e^8 + a^2*d*e^10)*x^2 + 3*(c^2*
d^6*e^5 + 2*a*c*d^4*e^7 + a^2*d^2*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 )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**4,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**4, x)

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Giac [B]  time = 1.57564, size = 795, normalized size = 3.98 \begin{align*} -c^{\frac{3}{2}} e^{\left (-4\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} 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 d^{2} e^{4} + a e^{6}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{54 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} c^{\frac{7}{2}} d^{4} e + 44 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c^{4} d^{5} + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{2} - 78 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a c^{\frac{7}{2}} d^{4} e - 34 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} + 27 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a c^{\frac{5}{2}} d^{2} e^{3} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{2} c^{\frac{5}{2}} d^{2} e^{3} - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a^{2} c^{\frac{3}{2}} e^{5} - 11 \, a^{3} c^{\frac{5}{2}} d^{2} e^{3} + 33 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{3} c^{2} d e^{4} + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{3} c^{\frac{3}{2}} e^{5} - 8 \, a^{4} c^{\frac{3}{2}} e^{5}}{3 \,{\left (c d^{2} e^{4} + a 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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-c^(3/2)*e^(-4)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + (2*c^3*d^3 + 3*a*c^2*d*e^2)*arctan(((sqrt(c)*x - sqrt
(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c*d^2*e^4 + a*e^6)*sqrt(-c*d^2 - a*e^2)) - 1/3*(54*(sqrt(c
)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d^5 + 18*(sqrt(c)*x - sqrt(c*x
^2 + a))^5*c^3*d^3*e^2 - 78*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e - 34*(sqrt(c)*x - sqrt(c*x^2 + a))
^3*a*c^3*d^3*e^2 + 27*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a
*c^2*d*e^4 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d^3*e^2 - 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)
*d^2*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*e
^5 - 11*a^3*c^(5/2)*d^2*e^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^4 + 12*(sqrt(c)*x - sqrt(c*x^2 + a)
)^2*a^3*c^(3/2)*e^5 - 8*a^4*c^(3/2)*e^5)/((c*d^2*e^4 + a*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)