\(\int \frac {(d+e x)^{5/2}}{(a-c x^2)^3} \, dx\) [639]

Optimal result

Integrand size = 20, antiderivative size = 279 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {(a e+c d x) (d+e x)^{3/2}}{4 a c \left (a-c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{16 a^2 c \left (a-c x^2\right )}-\frac {3 \sqrt {\sqrt {c} d-\sqrt {a} e} \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4}}+\frac {3 \sqrt {\sqrt {c} d+\sqrt {a} e} \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4}} \]

[Out]

1/4*(c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(-c*x^2+a)^2+3/16*(a*d*e+(-a*e^2+2*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c/(-c*x^2+a)-
3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d^2-a*e^2+2*d*e*a^(1/2)*c^(1/2))*(-e*a^(
1/2)+d*c^(1/2))^(1/2)/a^(5/2)/c^(7/4)+3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d^2
-a*e^2-2*d*e*a^(1/2)*c^(1/2))*(e*a^(1/2)+d*c^(1/2))^(1/2)/a^(5/2)/c^(7/4)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 279, 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, 835, 841, 1180, 214} \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=-\frac {3 \sqrt {\sqrt {c} d-\sqrt {a} e} \left (2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4}}+\frac {3 \sqrt {\sqrt {a} e+\sqrt {c} d} \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{7/4}}+\frac {3 \sqrt {d+e x} \left (x \left (2 c d^2-a e^2\right )+a d e\right )}{16 a^2 c \left (a-c x^2\right )}+\frac {(d+e x)^{3/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

[In]

Int[(d + e*x)^(5/2)/(a - c*x^2)^3,x]

[Out]

((a*e + c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a - c*x^2)^2) + (3*Sqrt[d + e*x]*(a*d*e + (2*c*d^2 - a*e^2)*x))/(16*a^
2*c*(a - c*x^2)) - (3*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4)) + (3*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(4*c*d^2 - 2
*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*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 835

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

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}}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {\sqrt {d+e x} \left (-\frac {3}{2} \left (2 c d^2-a e^2\right )-\frac {3}{2} c d e x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c} \\ & = \frac {(a e+c d x) (d+e x)^{3/2}}{4 a c \left (a-c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{16 a^2 c \left (a-c x^2\right )}+\frac {\int \frac {\frac {3}{4} c d \left (4 c d^2-3 a e^2\right )+\frac {3}{4} c e \left (2 c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2} \\ & = \frac {(a e+c d x) (d+e x)^{3/2}}{4 a c \left (a-c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{16 a^2 c \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {3}{4} c d e \left (4 c d^2-3 a e^2\right )-\frac {3}{4} c d e \left (2 c d^2-a e^2\right )+\frac {3}{4} c e \left (2 c d^2-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 )}{4 a^2 c^2} \\ & = \frac {(a e+c d x) (d+e x)^{3/2}}{4 a c \left (a-c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{16 a^2 c \left (a-c x^2\right )}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\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 )}{32 a^{5/2} c}-\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\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 )}{32 a^{5/2} c} \\ & = \frac {(a e+c d x) (d+e x)^{3/2}}{4 a c \left (a-c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{16 a^2 c \left (a-c x^2\right )}-\frac {3 \sqrt {\sqrt {c} d-\sqrt {a} e} \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4}}+\frac {3 \sqrt {\sqrt {c} d+\sqrt {a} e} \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{7/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.15 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} c \sqrt {d+e x} \left (6 c^2 d^2 x^3-a^2 e (7 d+e x)-a c x \left (10 d^2+d e x+3 e^2 x^2\right )\right )}{\left (a-c x^2\right )^2}-3 \sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+3 \sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} c^2} \]

[In]

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

[Out]

((-2*Sqrt[a]*c*Sqrt[d + e*x]*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) - a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)))/(a - c*
x^2)^2 - 3*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[-(c*d) - Sq
rt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)] + 3*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*(4*c*d^2 + 2*Sqr
t[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(3
2*a^(5/2)*c^2)

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {-\frac {9 e \left (-c \,x^{2}+a \right )^{2} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\frac {\left (-e^{2} a +2 c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{3}+c d \left (e^{2} a -\frac {4 c \,d^{2}}{3}\right )\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32}+\frac {7 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {9 e \left (-c \,x^{2}+a \right )^{2} \left (\frac {\left (e^{2} a -2 c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{3}+c d \left (e^{2} a -\frac {4 c \,d^{2}}{3}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{14}+\sqrt {a c \,e^{2}}\, \left (-\frac {6 c^{2} d^{2} x^{3}}{7}+\frac {10 \left (\frac {3}{10} x^{2} e^{2}+\frac {1}{10} d e x +d^{2}\right ) x a c}{7}+e \,a^{2} \left (\frac {e x}{7}+d \right )\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\right )}{16}}{a^{2} c \left (-c \,x^{2}+a \right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(329\)
default	\(2 e^{5} \left (\frac {\frac {3 \left (e^{2} a -2 c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{4}}-\frac {d \left (4 e^{2} a -9 c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{16 a^{2} e^{4}}+\frac {\left (a^{2} e^{4}+17 a c \,d^{2} e^{2}-18 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{4} c}+\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} d \sqrt {e x +d}}{16 a^{2} e^{4} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {-\frac {3 \left (3 d \,e^{2} a c -4 c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}-2 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{64 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {3 \left (-3 d \,e^{2} a c +4 c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}-2 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{64 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{a^{2} e^{4}}\right )\)	\(384\)
derivativedivides	\(-2 e^{5} \left (-\frac {\frac {3 \left (e^{2} a -2 c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{4}}-\frac {d \left (4 e^{2} a -9 c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{16 a^{2} e^{4}}+\frac {\left (a^{2} e^{4}+17 a c \,d^{2} e^{2}-18 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{4} c}+\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} d \sqrt {e x +d}}{16 a^{2} e^{4} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {3 \left (-\frac {\left (3 d \,e^{2} a c -4 c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}-2 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 d \,e^{2} a c +4 c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}-2 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 a^{2} e^{4}}\right )\)	\(385\)

[In]

int((e*x+d)^(5/2)/(-c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

7/16/((-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)*(-9/14*e*(-c*x^2+a)^2*((
c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(1/3*(-a*e^2+2*c*d^2)*(a*c*e^2)^(1/2)+c*d*(e^2*a-4/3*c*d^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)*(-9/14*e*(-c*x^2+a)^2*(1/3*(a*e^2-2*c*d
^2)*(a*c*e^2)^(1/2)+c*d*(e^2*a-4/3*c*d^2))*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(a*c*e^2)^
(1/2)*(-6/7*c^2*d^2*x^3+10/7*(3/10*x^2*e^2+1/10*d*e*x+d^2)*x*a*c+e*a^2*(1/7*e*x+d))*((c*d+(a*c*e^2)^(1/2))*c)^
(1/2)*(e*x+d)^(1/2)))/a^2/c/(-c*x^2+a)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1029 vs. \(2 (221) = 442\).

Time = 0.54 (sec) , antiderivative size = 1029, normalized size of antiderivative = 3.69 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} + 27 \, {\left (2 \, a^{3} c^{2} d e^{6} + {\left (4 \, a^{5} c^{6} d^{2} - a^{6} c^{5} e^{2}\right )} \sqrt {\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} - 27 \, {\left (2 \, a^{3} c^{2} d e^{6} + {\left (4 \, a^{5} c^{6} d^{2} - a^{6} c^{5} e^{2}\right )} \sqrt {\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} + 16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) + 3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} + 27 \, {\left (2 \, a^{3} c^{2} d e^{6} - {\left (4 \, a^{5} c^{6} d^{2} - a^{6} c^{5} e^{2}\right )} \sqrt {\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {e x + d} - 27 \, {\left (2 \, a^{3} c^{2} d e^{6} - {\left (4 \, a^{5} c^{6} d^{2} - a^{6} c^{5} e^{2}\right )} \sqrt {\frac {e^{10}}{a^{5} c^{7}}}\right )} \sqrt {-\frac {a^{5} c^{3} \sqrt {\frac {e^{10}}{a^{5} c^{7}}} - 16 \, c^{2} d^{5} + 20 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}}{a^{5} c^{3}}}\right ) + 4 \, {\left (a c d e x^{2} + 7 \, a^{2} d e - 3 \, {\left (2 \, c^{2} d^{2} - a c e^{2}\right )} x^{3} + {\left (10 \, a c d^{2} + a^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]

[In]

integrate((e*x+d)^(5/2)/(-c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(e^10/(a^5*c^7)) + 16*c^2*d^5 - 20*a*c*d^3*e^2
+ 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d*
e^6 + (4*a^5*c^6*d^2 - a^6*c^5*e^2)*sqrt(e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(e^10/(a^5*c^7)) + 16*c^2*d^5 - 20
*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(e^10/(a^5
*c^7)) + 16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e
^9)*sqrt(e*x + d) - 27*(2*a^3*c^2*d*e^6 + (4*a^5*c^6*d^2 - a^6*c^5*e^2)*sqrt(e^10/(a^5*c^7)))*sqrt((a^5*c^3*sq
rt(e^10/(a^5*c^7)) + 16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) + 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 +
a^4*c)*sqrt(-(a^5*c^3*sqrt(e^10/(a^5*c^7)) - 16*c^2*d^5 + 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16
*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d*e^6 - (4*a^5*c^6*d^2 - a^6*c^5*e^2)*s
qrt(e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(e^10/(a^5*c^7)) - 16*c^2*d^5 + 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3
))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(e^10/(a^5*c^7)) - 16*c^2*d^5 + 20*a*c*d^3*e^
2 - 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c^2*d
*e^6 - (4*a^5*c^6*d^2 - a^6*c^5*e^2)*sqrt(e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(e^10/(a^5*c^7)) - 16*c^2*d^5 +
20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) + 4*(a*c*d*e*x^2 + 7*a^2*d*e - 3*(2*c^2*d^2 - a*c*e^2)*x^3 + (10*a*c
*d^2 + a^2*e^2)*x)*sqrt(e*x + d))/(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(5/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\int { -\frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} - a\right )}^{3}} \,d x } \]

[In]

integrate((e*x+d)^(5/2)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

-integrate((e*x + d)^(5/2)/(c*x^2 - a)^3, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (221) = 442\).

Time = 0.42 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.14 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {3 \, {\left (4 \, c^{4} d^{4} e - 3 \, a c^{3} d^{2} e^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} c^{2} e^{2} - 2 \, {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} e - \sqrt {a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {3 \, {\left (4 \, c^{4} d^{4} e - 3 \, a c^{3} d^{2} e^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} c^{2} e^{2} + 2 \, {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} e + \sqrt {a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {e x + d} c^{2} d^{5} e - 3 \, {\left (e x + d\right )}^{\frac {7}{2}} a c e^{3} + 8 \, {\left (e x + d\right )}^{\frac {5}{2}} a c d e^{3} - 17 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} + 12 \, \sqrt {e x + d} a c d^{3} e^{3} - {\left (e x + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 6 \, \sqrt {e x + d} a^{2} d e^{5}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c} \]

[In]

integrate((e*x+d)^(5/2)/(-c*x^2+a)^3,x, algorithm="giac")

[Out]

3/32*(4*c^4*d^4*e - 3*a*c^3*d^2*e^3 - (2*a*c*d^2*e - a^2*e^3)*c^2*e^2 - 2*(sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c
*d*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^
2*c^2))/(a^2*c^2)))/((a^3*c^3*e - sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + 3/32*(4*c^4*d^4*
e - 3*a*c^3*d^2*e^3 - (2*a*c*d^2*e - a^2*e^3)*c^2*e^2 + 2*(sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c*d*e^3)*abs(c)*a
bs(e))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^2
)))/((a^3*c^3*e + sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 1/16*(6*(e*x + d)^(7/2)*c^2*d^2*
e - 18*(e*x + d)^(5/2)*c^2*d^3*e + 18*(e*x + d)^(3/2)*c^2*d^4*e - 6*sqrt(e*x + d)*c^2*d^5*e - 3*(e*x + d)^(7/2
)*a*c*e^3 + 8*(e*x + d)^(5/2)*a*c*d*e^3 - 17*(e*x + d)^(3/2)*a*c*d^2*e^3 + 12*sqrt(e*x + d)*a*c*d^3*e^3 - (e*x
+ d)^(3/2)*a^2*e^5 - 6*sqrt(e*x + d)*a^2*d*e^5)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)^2*a^2*c)

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 1015, normalized size of antiderivative = 3.64 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {\frac {3\,e\,\left (a\,e^2-2\,c\,d^2\right )\,{\left (d+e\,x\right )}^{7/2}}{16\,a^2}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^5+17\,a\,c\,d^2\,e^3-18\,c^2\,d^4\,e\right )}{16\,a^2\,c}-\frac {d\,\left (4\,a\,e^3-9\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{8\,a^2}+\frac {3\,\sqrt {d+e\,x}\,\left (a^2\,d\,e^5-2\,a\,c\,d^3\,e^3+c^2\,d^5\,e\right )}{8\,a^2\,c}}{c^2\,{\left (d+e\,x\right )}^4+a^2\,e^4+c^2\,d^4+\left (6\,c^2\,d^2-2\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^2-\left (4\,c^2\,d^3-4\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )-4\,c^2\,d\,{\left (d+e\,x\right )}^3-2\,a\,c\,d^2\,e^2}-2\,\mathrm {atanh}\left (\frac {9\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,d^5}{256\,a^5\,c}+\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,e^5\,\sqrt {a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,e^{11}}{2048\,a\,c^2}+\frac {27\,d^4\,e^7}{512\,a^3}-\frac {135\,d^2\,e^9}{2048\,a^2\,c}-\frac {27\,d\,e^{10}\,\sqrt {a^{15}\,c^7}}{1024\,a^9\,c^5}+\frac {27\,d^3\,e^8\,\sqrt {a^{15}\,c^7}}{1024\,a^{10}\,c^4}\right )}+\frac {9\,d\,e^7\,\sqrt {a^{15}\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,d^5}{256\,a^5\,c}+\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}-\frac {9\,e^5\,\sqrt {a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,a^7\,c\,e^{11}}{2048}+\frac {27\,a^5\,c^3\,d^4\,e^7}{512}-\frac {135\,a^6\,c^2\,d^2\,e^9}{2048}-\frac {27\,d\,e^{10}\,\sqrt {a^{15}\,c^7}}{1024\,a\,c^2}+\frac {27\,d^3\,e^8\,\sqrt {a^{15}\,c^7}}{1024\,a^2\,c}\right )}\right )\,\sqrt {-\frac {9\,\left (e^5\,\sqrt {a^{15}\,c^7}-16\,a^5\,c^6\,d^5-5\,a^7\,c^4\,d\,e^4+20\,a^6\,c^5\,d^3\,e^2\right )}{4096\,a^{10}\,c^7}}-2\,\mathrm {atanh}\left (\frac {9\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,d^5}{256\,a^5\,c}+\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}+\frac {9\,e^5\,\sqrt {a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,e^{11}}{2048\,a\,c^2}+\frac {27\,d^4\,e^7}{512\,a^3}-\frac {135\,d^2\,e^9}{2048\,a^2\,c}+\frac {27\,d\,e^{10}\,\sqrt {a^{15}\,c^7}}{1024\,a^9\,c^5}-\frac {27\,d^3\,e^8\,\sqrt {a^{15}\,c^7}}{1024\,a^{10}\,c^4}\right )}-\frac {9\,d\,e^7\,\sqrt {a^{15}\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {9\,d^5}{256\,a^5\,c}+\frac {45\,d\,e^4}{4096\,a^3\,c^3}-\frac {45\,d^3\,e^2}{1024\,a^4\,c^2}+\frac {9\,e^5\,\sqrt {a^{15}\,c^7}}{4096\,a^{10}\,c^7}}}{32\,\left (\frac {27\,a^7\,c\,e^{11}}{2048}+\frac {27\,a^5\,c^3\,d^4\,e^7}{512}-\frac {135\,a^6\,c^2\,d^2\,e^9}{2048}+\frac {27\,d\,e^{10}\,\sqrt {a^{15}\,c^7}}{1024\,a\,c^2}-\frac {27\,d^3\,e^8\,\sqrt {a^{15}\,c^7}}{1024\,a^2\,c}\right )}\right )\,\sqrt {\frac {9\,\left (e^5\,\sqrt {a^{15}\,c^7}+16\,a^5\,c^6\,d^5+5\,a^7\,c^4\,d\,e^4-20\,a^6\,c^5\,d^3\,e^2\right )}{4096\,a^{10}\,c^7}} \]

[In]

int((d + e*x)^(5/2)/(a - c*x^2)^3,x)

[Out]

((3*e*(a*e^2 - 2*c*d^2)*(d + e*x)^(7/2))/(16*a^2) + ((d + e*x)^(3/2)*(a^2*e^5 - 18*c^2*d^4*e + 17*a*c*d^2*e^3)
)/(16*a^2*c) - (d*(4*a*e^3 - 9*c*d^2*e)*(d + e*x)^(5/2))/(8*a^2) + (3*(d + e*x)^(1/2)*(a^2*d*e^5 + c^2*d^5*e -
2*a*c*d^3*e^3))/(8*a^2*c))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^
2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2) - 2*atanh((9*e^8*(d + e*x)^(1/2)*((9*d^5
)/(256*a^5*c) + (45*d*e^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(1024*a^4*c^2) - (9*e^5*(a^15*c^7)^(1/2))/(4096*a^10*
c^7))^(1/2))/(32*((27*e^11)/(2048*a*c^2) + (27*d^4*e^7)/(512*a^3) - (135*d^2*e^9)/(2048*a^2*c) - (27*d*e^10*(a
^15*c^7)^(1/2))/(1024*a^9*c^5) + (27*d^3*e^8*(a^15*c^7)^(1/2))/(1024*a^10*c^4))) + (9*d*e^7*(a^15*c^7)^(1/2)*(
d + e*x)^(1/2)*((9*d^5)/(256*a^5*c) + (45*d*e^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(1024*a^4*c^2) - (9*e^5*(a^15*c
^7)^(1/2))/(4096*a^10*c^7))^(1/2))/(32*((27*a^7*c*e^11)/2048 + (27*a^5*c^3*d^4*e^7)/512 - (135*a^6*c^2*d^2*e^9
)/2048 - (27*d*e^10*(a^15*c^7)^(1/2))/(1024*a*c^2) + (27*d^3*e^8*(a^15*c^7)^(1/2))/(1024*a^2*c))))*(-(9*(e^5*(
a^15*c^7)^(1/2) - 16*a^5*c^6*d^5 - 5*a^7*c^4*d*e^4 + 20*a^6*c^5*d^3*e^2))/(4096*a^10*c^7))^(1/2) - 2*atanh((9*
e^8*(d + e*x)^(1/2)*((9*d^5)/(256*a^5*c) + (45*d*e^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(1024*a^4*c^2) + (9*e^5*(a
^15*c^7)^(1/2))/(4096*a^10*c^7))^(1/2))/(32*((27*e^11)/(2048*a*c^2) + (27*d^4*e^7)/(512*a^3) - (135*d^2*e^9)/(
2048*a^2*c) + (27*d*e^10*(a^15*c^7)^(1/2))/(1024*a^9*c^5) - (27*d^3*e^8*(a^15*c^7)^(1/2))/(1024*a^10*c^4))) -
(9*d*e^7*(a^15*c^7)^(1/2)*(d + e*x)^(1/2)*((9*d^5)/(256*a^5*c) + (45*d*e^4)/(4096*a^3*c^3) - (45*d^3*e^2)/(102
4*a^4*c^2) + (9*e^5*(a^15*c^7)^(1/2))/(4096*a^10*c^7))^(1/2))/(32*((27*a^7*c*e^11)/2048 + (27*a^5*c^3*d^4*e^7)
/512 - (135*a^6*c^2*d^2*e^9)/2048 + (27*d*e^10*(a^15*c^7)^(1/2))/(1024*a*c^2) - (27*d^3*e^8*(a^15*c^7)^(1/2))/
(1024*a^2*c))))*((9*(e^5*(a^15*c^7)^(1/2) + 16*a^5*c^6*d^5 + 5*a^7*c^4*d*e^4 - 20*a^6*c^5*d^3*e^2))/(4096*a^10
*c^7))^(1/2)