\(\int \frac {(d+e x)^{5/2}}{(a-c x^2)^2} \, dx\) [625]
Optimal result
Integrand size = 20, antiderivative size = 231 \[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}
\]
[Out]
1/2*(c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(-c*x^2+a)+1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(e
*a^(1/2)+d*c^(1/2))^(3/2)*(-3*e*a^(1/2)+2*d*c^(1/2))/a^(3/2)/c^(7/4)-1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(
1/2)+d*c^(1/2))^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(3/2)*(3*e*a^(1/2)+2*d*c^(1/2))/a^(3/2)/c^(7/4)+1/2*d*e*(e*x+d)^
(1/2)/a/c
Rubi [A] (verified)
Time = 0.25 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {753, 839, 841, 1180, 214}
\[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=-\frac {\left (3 \sqrt {a} e+2 \sqrt {c} d\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{7/4}}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac {d e \sqrt {d+e x}}{2 a c}
\]
[In]
Int[(d + e*x)^(5/2)/(a - c*x^2)^2,x]
[Out]
(d*e*Sqrt[d + e*x])/(2*a*c) + ((a*e + c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(
3/2)*(2*Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(7
/4)) + ((2*Sqrt[c]*d - 3*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]
*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4))
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 753
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 839
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(
c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]
Rule 841
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]
Rule 1180
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps \begin{align*}
\text {integral}& = \frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (-2 c d^2+3 a e^2\right )+\frac {1}{2} c d e x\right )}{a-c x^2} \, dx}{2 a c} \\ & = \frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}+\frac {\int \frac {c d \left (c d^2-2 a e^2\right )+\frac {1}{2} c e \left (c d^2-3 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c^2} \\ & = \frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} c d e \left (c d^2-3 a e^2\right )+c d e \left (c d^2-2 a e^2\right )+\frac {1}{2} c e \left (c d^2-3 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c^2} \\ & = \frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}+\frac {\left (\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^2\right ) \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (2 \sqrt {c} d+3 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c} \\ & = \frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.69 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.11
\[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} c \sqrt {d+e x} \left (c d^2 x+a e (2 d+e x)\right )}{-a+c x^2}-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 c d^2-\sqrt {a} \sqrt {c} d e-3 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (2 c d^2+\sqrt {a} \sqrt {c} d e-3 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^2}
\]
[In]
Integrate[(d + e*x)^(5/2)/(a - c*x^2)^2,x]
[Out]
((-2*Sqrt[a]*c*Sqrt[d + e*x]*(c*d^2*x + a*e*(2*d + e*x)))/(-a + c*x^2) - Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*(2*c
*d^2 - Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqr
t[a]*e)] + Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*(2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[(Sqrt[-(c*d) + Sq
rt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(4*a^(3/2)*c^2)
Maple [A] (verified)
Time = 2.44 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.28
| | |
| method | result | size |
| | |
| pseudoelliptic |
\(\frac {-\left (-c \,x^{2}+a \right ) \left (\frac {\left (-3 e^{2} a +c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{4}+c d \left (e^{2} a -\frac {c \,d^{2}}{2}\right )\right ) e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\left (-c \,x^{2}+a \right ) \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{4}+c d \left (e^{2} a -\frac {c \,d^{2}}{2}\right )\right ) e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (\frac {c \,d^{2} x}{2}+a e \left (\frac {e x}{2}+d \right )\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a c \left (-c \,x^{2}+a \right )}\) |
\(296\) |
| derivativedivides |
\(2 e^{3} \left (\frac {\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) d \sqrt {e x +d}}{4 a c \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {-\frac {\left (4 d \,e^{2} a c -2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 d \,e^{2} a c +2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) |
\(313\) |
| default |
\(2 e^{3} \left (\frac {\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) d \sqrt {e x +d}}{4 a c \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {-\frac {\left (4 d \,e^{2} a c -2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 d \,e^{2} a c +2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) |
\(313\) |
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[In]
int((e*x+d)^(5/2)/(-c*x^2+a)^2,x,method=_RETURNVERBOSE)
[Out]
1/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c*e^2)^(1/2)*(-(-c*x^2+a)*(1/4*(-3*a*e^2
+c*d^2)*(a*c*e^2)^(1/2)+c*d*(e^2*a-1/2*c*d^2))*e*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d
+(a*c*e^2)^(1/2))*c)^(1/2))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-(-c*x^2+a)*(1/4*(3*a*e^2-c*d^2)*(a*c*e^2)^(1/2)
+c*d*(e^2*a-1/2*c*d^2))*e*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(1/2*c*d^2*x+a*e*(1/2*e*x+d
))*(a*c*e^2)^(1/2)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)))/a/c/(-c*x^2+a)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (172) = 344\).
Time = 0.35 (sec) , antiderivative size = 1370, normalized size of antiderivative = 5.93
\[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="fricas")
[Out]
-1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*
a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7
- 81*a^3*e^9)*sqrt(e*x + d) + (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25
*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3
*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 - a^2*c)*sqrt((
4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*
c^7)))/(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) - (
5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 +
81*a^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a
*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) + (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*
a^2*d*e^4 - a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log(-(20*c^3*d
^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) + (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*
e^6 + (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt((4
*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c
^7)))/(a^3*c^3))) - (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt((25*c^2
*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162
*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 + (2*a^3*c^6*d^2 - 3*a^4*c^5
*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a
^2*d*e^4 - a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) + 4*(2*a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^2*x^2 - a^2*c)
Sympy [F(-1)]
Timed out. \[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out}
\]
[In]
integrate((e*x+d)**(5/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} - a\right )}^{2}} \,d x }
\]
[In]
integrate((e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate((e*x + d)^(5/2)/(c*x^2 - a)^2, x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (172) = 344\).
Time = 0.38 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.19
\[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {{\left (2 \, a c^{4} d^{4} e - 4 \, a^{2} c^{3} d^{2} e^{3} - {\left (c d^{2} e - 3 \, a e^{3}\right )} a^{2} c^{2} e^{2} - {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (2 \, \sqrt {a c} a c^{4} d^{4} e - 4 \, \sqrt {a c} a^{2} c^{3} d^{2} e^{3} - {\left (\sqrt {a c} c d^{2} e - 3 \, \sqrt {a c} a e^{3}\right )} a^{2} c^{2} e^{2} + {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{4} d + \sqrt {a c} a^{2} c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} c d^{2} e - \sqrt {e x + d} c d^{3} e + {\left (e x + d\right )}^{\frac {3}{2}} a e^{3} + \sqrt {e x + d} a d e^{3}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c}
\]
[In]
integrate((e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="giac")
[Out]
1/4*(2*a*c^4*d^4*e - 4*a^2*c^3*d^2*e^3 - (c*d^2*e - 3*a*e^3)*a^2*c^2*e^2 - (sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*
c*d*e^3)*abs(a)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2
)*a*c^2))/(a*c^2)))/((a^2*c^3*e - sqrt(a*c)*a*c^3*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a)*abs(e)) + 1/4*(2*sqrt
(a*c)*a*c^4*d^4*e - 4*sqrt(a*c)*a^2*c^3*d^2*e^3 - (sqrt(a*c)*c*d^2*e - 3*sqrt(a*c)*a*e^3)*a^2*c^2*e^2 + (a*c^3
*d^3*e - a^2*c^2*d*e^3)*abs(a)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(a*c^2*d - sqrt(a^2*c^4*d^2 - (a*c^2*
d^2 - a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c^4*d + sqrt(a*c)*a^2*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a)*abs(
e)) - 1/2*((e*x + d)^(3/2)*c*d^2*e - sqrt(e*x + d)*c*d^3*e + (e*x + d)^(3/2)*a*e^3 + sqrt(e*x + d)*a*d*e^3)/((
(e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)*a*c)
Mupad [B] (verification not implemented)
Time = 9.98 (sec) , antiderivative size = 1988, normalized size of antiderivative = 8.61
\[
\int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
int((d + e*x)^(5/2)/(a - c*x^2)^2,x)
[Out]
2*atanh((18*a*e^8*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5
*(a^9*c^7)^(1/2))/(64*a^5*c^7) + (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((15*d^2*e^9)/c - (43*d^4*e^
7)/(4*a) - (27*a*e^11)/(4*c^2) + (5*c*d^6*e^5)/(2*a^2) + (9*d*e^10*(a^9*c^7)^(1/2))/(4*a^4*c^5) - (7*d^3*e^8*(
a^9*c^7)^(1/2))/(2*a^5*c^4) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^6*c^3)) - (10*c*d^2*e^6*(d + e*x)^(1/2)*(d^5/(1
6*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) + (5*d^2*e
^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((15*d^2*e^9)/c - (43*d^4*e^7)/(4*a) - (27*a*e^11)/(4*c^2) + (5*c*d^6
*e^5)/(2*a^2) + (9*d*e^10*(a^9*c^7)^(1/2))/(4*a^4*c^5) - (7*d^3*e^8*(a^9*c^7)^(1/2))/(2*a^5*c^4) + (5*d^5*e^6*
(a^9*c^7)^(1/2))/(4*a^6*c^3)) + (18*d*e^7*(a^9*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c
^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) + (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c
^6))^(1/2))/((5*a^2*c^4*d^6*e^5)/2 - (27*a^5*c*e^11)/4 - (43*a^3*c^3*d^4*e^7)/4 + 15*a^4*c^2*d^2*e^9 + (9*d*e^
10*(a^9*c^7)^(1/2))/(4*c^2) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^2) - (7*d^3*e^8*(a^9*c^7)^(1/2))/(2*a*c)) - (10
*d^3*e^5*(a^9*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) -
(9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) + (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/(15*a^5*c*d^2*e^9 - (
27*a^6*e^11)/4 + (5*a^3*c^3*d^6*e^5)/2 - (43*a^4*c^2*d^4*e^7)/4 - (7*d^3*e^8*(a^9*c^7)^(1/2))/(2*c^2) + (5*d^5
*e^6*(a^9*c^7)^(1/2))/(4*a*c) + (9*a*d*e^10*(a^9*c^7)^(1/2))/(4*c^3)))*((4*a^3*c^6*d^5 - 9*a*e^5*(a^9*c^7)^(1/
2) + 15*a^5*c^4*d*e^4 - 15*a^4*c^5*d^3*e^2 + 5*c*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) - (((a*e^3 + c*d
^2*e)*(d + e*x)^(3/2))/(2*a*c) + ((a*d*e^3 - c*d^3*e)*(d + e*x)^(1/2))/(2*a*c))/(c*(d + e*x)^2 - a*e^2 + c*d^2
- 2*c*d*(d + e*x)) - 2*atanh((18*a*e^8*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)
/(64*a^2*c^2) + (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) - (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a
*e^11)/(4*c^2) + (43*d^4*e^7)/(4*a) - (15*d^2*e^9)/c - (5*c*d^6*e^5)/(2*a^2) + (9*d*e^10*(a^9*c^7)^(1/2))/(4*a
^4*c^5) - (7*d^3*e^8*(a^9*c^7)^(1/2))/(2*a^5*c^4) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^6*c^3)) - (10*c*d^2*e^6*(
d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) + (9*e^5*(a^9*c^7)^(1/2))/(
64*a^5*c^7) - (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a*e^11)/(4*c^2) + (43*d^4*e^7)/(4*a) - (15
*d^2*e^9)/c - (5*c*d^6*e^5)/(2*a^2) + (9*d*e^10*(a^9*c^7)^(1/2))/(4*a^4*c^5) - (7*d^3*e^8*(a^9*c^7)^(1/2))/(2*
a^5*c^4) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^6*c^3)) - (18*d*e^7*(a^9*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(16*a^3*c
) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) + (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) - (5*d^2*e^3*(a^9
*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((27*a^5*c*e^11)/4 - (5*a^2*c^4*d^6*e^5)/2 + (43*a^3*c^3*d^4*e^7)/4 - 15*a^4
*c^2*d^2*e^9 + (9*d*e^10*(a^9*c^7)^(1/2))/(4*c^2) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^2) - (7*d^3*e^8*(a^9*c^7)
^(1/2))/(2*a*c)) + (10*d^3*e^5*(a^9*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d
^3*e^2)/(64*a^2*c^2) + (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) - (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))
/((27*a^6*e^11)/4 - 15*a^5*c*d^2*e^9 - (5*a^3*c^3*d^6*e^5)/2 + (43*a^4*c^2*d^4*e^7)/4 - (7*d^3*e^8*(a^9*c^7)^(
1/2))/(2*c^2) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a*c) + (9*a*d*e^10*(a^9*c^7)^(1/2))/(4*c^3)))*((4*a^3*c^6*d^5 +
9*a*e^5*(a^9*c^7)^(1/2) + 15*a^5*c^4*d*e^4 - 15*a^4*c^5*d^3*e^2 - 5*c*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^7))^
(1/2)