### 3.542 $$\int \frac{(a+c x^2)^{3/2}}{(d+e x)^5} \, dx$$

Optimal. Leaf size=153 $-\frac{3 a^2 c^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac{3 a c \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}$

[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))

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Rubi [A]  time = 0.0628769, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.158, Rules used = {721, 725, 206} $-\frac{3 a^2 c^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac{3 a c \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[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 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)^5} \, dx &=-\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 ) \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 )^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 [A]  time = 0.277666, size = 198, normalized size = 1.29 $\frac{1}{8} \left (\frac{\sqrt{a+c x^2} \left (-a^2 c e \left (5 d^2+4 d e x+5 e^2 x^2\right )-2 a^3 e^3+a c^2 d x \left (5 d^2+4 d e x+5 e^2 x^2\right )+2 c^3 d^3 x^3\right )}{(d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{3 a^2 c^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{5/2}}+\frac{3 a^2 c^2 \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.24, size = 3528, normalized size = 23.1 \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)^5,x)

[Out]

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

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 20.2292, size = 2221, normalized size = 14.52 \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)^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)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 4.53682, size = 1278, normalized size = 8.35 \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)^5,x, algorithm="giac")

[Out]

-1/64*(3*sqrt(c*d^2 + a*e^2)*a^2*c^2*e^14*log(abs(sqrt(c*d^2 + a*e^2)*c*d - (c*d^2 + a*e^2)*(sqrt(c - 2*c*d/(x
*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 + a*e^4)*e^(-1)/(x*e + d))))*sgn(1/(x*e + d)
)/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) - sqrt(c - 2*c*d/(x*e + d) + c
*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)*((2*(3*(c^4*d^7*e^19*sgn(1/(x*e + d)) + 3*a*c^3*d^5*e^21*sgn(1/(x*e + d)
) + 3*a^2*c^2*d^3*e^23*sgn(1/(x*e + d)) + a^3*c*d*e^25*sgn(1/(x*e + d)))/(c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a
^2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 + a^4*e^16) - (c^4*d^8*e^20*sgn(1/(x*e + d)) + 4*a*c^3*d^6*e^22*sgn(1/(x*e
+ d)) + 6*a^2*c^2*d^4*e^24*sgn(1/(x*e + d)) + 4*a^3*c*d^2*e^26*sgn(1/(x*e + d)) + a^4*e^28*sgn(1/(x*e + d)))*e
^(-1)/((c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 + a^4*e^16)*(x*e + d)))*e^(-1)/
(x*e + d) - (6*c^4*d^6*e^18*sgn(1/(x*e + d)) + 17*a*c^3*d^4*e^20*sgn(1/(x*e + d)) + 16*a^2*c^2*d^2*e^22*sgn(1/
(x*e + d)) + 5*a^3*c*e^24*sgn(1/(x*e + d)))/(c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2
*e^14 + a^4*e^16))*e^(-1)/(x*e + d) + (2*c^4*d^5*e^17*sgn(1/(x*e + d)) + 7*a*c^3*d^3*e^19*sgn(1/(x*e + d)) + 5
*a^2*c^2*d*e^21*sgn(1/(x*e + d)))/(c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 + a^
4*e^16)) + (2*c^(9/2)*d^5*e^9 + 7*a*c^(7/2)*d^3*e^11 + 5*a^2*c^(5/2)*d*e^13 - 3*sqrt(c*d^2 + a*e^2)*a^2*c^2*e^
13*log(abs(-c^(3/2)*d^2 + sqrt(c*d^2 + a*e^2)*c*d - a*sqrt(c)*e^2)))*sgn(1/(x*e + d))/(c^4*d^8 + 4*a*c^3*d^6*e
^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8))*e^2