\(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^4} \, dx\) [541]
Optimal result
Integrand size = 19, antiderivative size = 200 \[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {c \left (d \left (2 c d^2+a e^2\right )+e \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{2 e^3 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}+\frac {c^2 d \left (2 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 e^4 \left (c d^2+a e^2\right )^{3/2}}
\]
[Out]
-1/3*(c*x^2+a)^(3/2)/e/(e*x+d)^3+c^(3/2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/e^4+1/2*c^2*d*(3*a*e^2+2*c*d^2)*ar
ctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^4/(a*e^2+c*d^2)^(3/2)-1/2*c*(d*(a*e^2+2*c*d^2)+e*(2*
a*e^2+3*c*d^2)*x)*(c*x^2+a)^(1/2)/e^3/(a*e^2+c*d^2)/(e*x+d)^2
Rubi [A] (verified)
Time = 0.13 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {747, 825, 858, 223, 212, 739}
\[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^4}+\frac {c^2 d \left (3 a e^2+2 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac {c \sqrt {a+c x^2} \left (e x \left (2 a e^2+3 c d^2\right )+d \left (a e^2+2 c d^2\right )\right )}{2 e^3 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3}
\]
[In]
Int[(a + c*x^2)^(3/2)/(d + e*x)^4,x]
[Out]
-1/2*(c*(d*(2*c*d^2 + a*e^2) + e*(3*c*d^2 + 2*a*e^2)*x)*Sqrt[a + c*x^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 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 223
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 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]
Rule 747
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 825
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/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((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), 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 858
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]
Rubi steps \begin{align*}
\text {integral}& = -\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 \text {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 ) \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 )}{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] (verified)
Time = 2.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.05
\[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {\frac {e \sqrt {a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+9 d e x+8 e^2 x^2\right )+c^2 d^2 \left (6 d^2+15 d e x+11 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right ) (d+e x)^3}+\frac {6 c^2 d \left (2 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+6 c^{3/2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 e^4}
\]
[In]
Integrate[(a + c*x^2)^(3/2)/(d + e*x)^4,x]
[Out]
-1/6*((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) + (6*c^2*d*(2*c*d^2 + 3*a*e^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)^(3/2) + 6*c^(3/2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/e^4
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(2445\) vs. \(2(178)=356\).
Time = 2.05 (sec) , antiderivative size = 2446, normalized size of antiderivative =
12.23
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method | result | size |
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default |
\(\text {Expression too large to display}\) |
\(2446\) |
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[In]
int((c*x^2+a)^(3/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)
[Out]
1/e^4*(-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))))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (179) = 358\).
Time = 4.00 (sec) , antiderivative size = 2513, normalized size of antiderivative = 12.56
\[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display}
\]
[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)]
Sympy [F]
\[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx
\]
[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)
Maxima [F(-2)]
Exception generated. \[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate((c*x^2+a)^(3/2)/(e*x+d)^4,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 [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (179) = 358\).
Time = 0.33 (sec) , antiderivative size = 604, normalized size of antiderivative = 3.02
\[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} e^{4} + a e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{e^{4}} - \frac {18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{2} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 54 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} - 34 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} - 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} d^{2} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{5} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} - 11 \, a^{3} c^{\frac {5}{2}} d^{2} e^{3} - 8 \, a^{4} c^{\frac {3}{2}} e^{5}}{3 \, {\left (c d^{2} e^{4} + a e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}}
\]
[In]
integrate((c*x^2+a)^(3/2)/(e*x+d)^4,x, algorithm="giac")
[Out]
(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)) - c^(3/2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/e^4 - 1/3*(18*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*c^3*d^3*e^2 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*e^4 + 54*(sqrt(c)*x - sqrt(c*x^2
+ a))^4*c^(7/2)*d^4*e + 27*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a
))^4*a^2*c^(3/2)*e^5 + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d^5 - 34*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d
^3*e^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*e^4 - 78*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e
- 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*e^5
+ 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d^3*e^2 + 33*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^4 - 11*a^3*
c^(5/2)*d^2*e^3 - 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)
Mupad [F(-1)]
Timed out. \[
\int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x
\]
[In]
int((a + c*x^2)^(3/2)/(d + e*x)^4,x)
[Out]
int((a + c*x^2)^(3/2)/(d + e*x)^4, x)