\(\int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx\) [612]

Optimal result

Integrand size = 20, antiderivative size = 167 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 167, 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 = {718, 839, 841, 1180, 214} \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{7/4}}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {4 d e \sqrt {d+e x}}{c} \]

[In]

Int[(d + e*x)^(5/2)/(a - c*x^2),x]

[Out]

(-4*d*e*Sqrt[d + e*x])/c - (2*e*(d + e*x)^(3/2))/(3*c) - ((Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[
d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(7/4)) + ((Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sq
rt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*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 718

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(m - 1))), x] +
Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + 2*c*d*e*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e}
, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 1]

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {2 c e \sqrt {d+e x} (7 d+e x)+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a}}+\frac {3 \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{3 c^2} \]

[In]

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

[Out]

-1/3*(2*c*e*Sqrt[d + e*x]*(7*d + e*x) + (3*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*ArcTan[(
Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqrt[a] + (3*Sqrt[c]*(Sqrt[c]*d - Sq
rt[a]*e)^3*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c
*d) + Sqrt[a]*Sqrt[c]*e]))/c^2

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.29

method	result	size
pseudoelliptic	\(-\frac {e \left (\frac {2 \sqrt {e x +d}\, \left (e x +7 d \right )}{3}+\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c}\)	\(216\)
risch	\(-\frac {2 \left (e x +7 d \right ) \sqrt {e x +d}\, e}{3 c}-2 e \left (\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(225\)
derivativedivides	\(-2 e \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+2 d \sqrt {e x +d}}{c}-\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(228\)
default	\(2 e \left (-\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+2 d \sqrt {e x +d}}{c}-\frac {\left (-3 d \,e^{2} a c -c^{2} d^{3}-\sqrt {a c \,e^{2}}\, a \,e^{2}-3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (3 d \,e^{2} a c +c^{2} d^{3}-\sqrt {a c \,e^{2}}\, a \,e^{2}-3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(231\)

[In]

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

[Out]

-e/c*(2/3*(e*x+d)^(1/2)*(e*x+7*d)+(-3*d*e^2*a*c-c^2*d^3+(a*c*e^2)^(1/2)*a*e^2+3*(a*c*e^2)^(1/2)*c*d^2)/(a*c*e^
2)^(1/2)/((-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))-(3*d*e^2*a*
c+c^2*d^3+(a*c*e^2)^(1/2)*a*e^2+3*(a*c*e^2)^(1/2)*c*d^2)/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arcta
nh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (121) = 242\).

Time = 0.33 (sec) , antiderivative size = 1617, normalized size of antiderivative = 9.68 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*(3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*
a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^
3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 +
a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a
*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^
2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*
d*e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a
*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) - (10*a*c^4*
d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^
6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*
e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c
^7)))/(a*c^3))) + 3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^
6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^
4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 +
(a*c^6*d^2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 +
a^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*
e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt((c^2*d^5 + 10*a*c*d^3*
e^2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 +
a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d)
- (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 +
100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^
2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^
4*e^10)/(a*c^7)))/(a*c^3))) + 4*(e^2*x + 7*d*e)*sqrt(e*x + d))/c

Sympy [F]

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

[In]

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

[Out]

-Integral(d**2*sqrt(d + e*x)/(-a + c*x**2), x) - Integral(e**2*x**2*sqrt(d + e*x)/(-a + c*x**2), x) - Integral
(2*d*e*x*sqrt(d + e*x)/(-a + c*x**2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (121) = 242\).

Time = 0.33 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {{\left (\sqrt {a c} c^{4} d^{4} e + 3 \, \sqrt {a c} a c^{3} d^{2} e^{3} - {\left (3 \, \sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} c^{2} e^{2} + 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} c^{4} d^{4} e + 3 \, \sqrt {a c} a c^{3} d^{2} e^{3} - {\left (3 \, \sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} c^{2} e^{2} - 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {e x + d} c^{2} d e\right )}}{3 \, c^{3}} \]

[In]

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

[Out]

-(sqrt(a*c)*c^4*d^4*e + 3*sqrt(a*c)*a*c^3*d^2*e^3 - (3*sqrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*c^2*e^2 + 2*(a
*c^3*d^3*e - a^2*c^2*d*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d + sqrt(c^8*d^2 - (c^4*d^2 - a*c^3
*e^2)*c^4))/c^4))/((a*c^4*d - sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + (sqrt(a*c)*c^4*d^4*e +
3*sqrt(a*c)*a*c^3*d^2*e^3 - (3*sqrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*c^2*e^2 - 2*(a*c^3*d^3*e - a^2*c^2*d*
e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c^4))/((a*c
^4*d + sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 2/3*((e*x + d)^(3/2)*c^2*e + 6*sqrt(e*x + d)*
c^2*d*e)/c^3

Mupad [B] (verification not implemented)

Time = 9.80 (sec) , antiderivative size = 3385, normalized size of antiderivative = 20.27 \[ \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(5/2)/(a - c*x^2),x)

[Out]

- atan((a^3*e^8*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^
4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/((16*a^
4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 + (160*a^3*d^2*e^9)/c - 160*a*c*d^6*e^5 - (160*d^5*e^6*(a^3*c^7)
^(1/2))/c^3 + (288*a*d^3*e^8*(a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(a^3*c^
7)^(1/2))/(a*c^2)) - (d^5*e^3*(a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c) + (
5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))
/(2*a*c^6))^(1/2)*160i)/((16*a^5*e^11)/c + 160*a^4*d^2*e^9 - 80*a*c^3*d^8*e^3 + 64*a^3*c*d^4*e^7 - 160*a^2*c^2
*d^6*e^5 - (160*d^7*e^4*(a^3*c^7)^(1/2))/c - (160*a*d^5*e^6*(a^3*c^7)^(1/2))/c^2 + (32*a^3*d*e^10*(a^3*c^7)^(1
/2))/c^4 + (288*a^2*d^3*e^8*(a^3*c^7)^(1/2))/c^3) - (d^3*e^5*(a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(
1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5
) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/(16*a^4*e^11 - 80*c^4*d^8*e^3 - 160*a*c^3*d^6*e^5 + 160
*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 - (160*d^7*e^4*(a^3*c^7)^(1/2))/a - (160*d^5*e^6*(a^3*c^7)^(1/2))/c + (288
*a*d^3*e^8*(a^3*c^7)^(1/2))/c^2 + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^3) - (a*d*e^7*(a^3*c^7)^(1/2)*(d + e*x)^(1
/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(a^3*c
^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/(16*a^4*c*e^11 - 160*d^5*e^6*(a^3*c
^7)^(1/2) - 80*c^5*d^8*e^3 - 160*a*c^4*d^6*e^5 + 64*a^2*c^3*d^4*e^7 + 160*a^3*c^2*d^2*e^9 + (288*a*d^3*e^8*(a^
3*c^7)^(1/2))/c - (160*c*d^7*e^4*(a^3*c^7)^(1/2))/a + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^2) + (a*c^2*d^4*e^4*(d
+ e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) + (5*d^
4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^4*e^11)/c^2 + 64*
a^2*d^4*e^7 - 80*c^2*d^8*e^3 + (160*a^3*d^2*e^9)/c - 160*a*c*d^6*e^5 - (160*d^5*e^6*(a^3*c^7)^(1/2))/c^3 + (28
8*a*d^3*e^8*(a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(a^3*c^7)^(1/2))/(a*c^2)
) + (a^2*c*d^2*e^6*(d + e*x)^(1/2)*((e^5*(a^3*c^7)^(1/2))/(4*c^7) + d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) + (5*a*d
*e^4)/(4*c^3) + (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/((1
6*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 + (160*a^3*d^2*e^9)/c - 160*a*c*d^6*e^5 - (160*d^5*e^6*(a^3*
c^7)^(1/2))/c^3 + (288*a*d^3*e^8*(a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(a^
3*c^7)^(1/2))/(a*c^2)))*((a^2*e^5*(a^3*c^7)^(1/2) + a*c^6*d^5 + 5*a^3*c^4*d*e^4 + 10*a^2*c^5*d^3*e^2 + 5*c^2*d
^4*e*(a^3*c^7)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*2i - atan((a^3*e^8*(d + e*x)^(1/2)*(
d^5/(4*a*c) - (e^5*(a^3*c^7)^(1/2))/(4*c^7) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(a^3*c^7)^(
1/2))/(4*a^2*c^5) - (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80
*c^2*d^8*e^3 + (160*a^3*d^2*e^9)/c - 160*a*c*d^6*e^5 + (160*d^5*e^6*(a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(a^3
*c^7)^(1/2))/c^4 - (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(a^3*c^7)^(1/2))/(a*c^2)) + (d^5*e^3*(a^
3*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(4*a*c) - (e^5*(a^3*c^7)^(1/2))/(4*c^7) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/
(4*c^3) - (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^5*
e^11)/c + 160*a^4*d^2*e^9 - 80*a*c^3*d^8*e^3 + 64*a^3*c*d^4*e^7 - 160*a^2*c^2*d^6*e^5 + (160*d^7*e^4*(a^3*c^7)
^(1/2))/c + (160*a*d^5*e^6*(a^3*c^7)^(1/2))/c^2 - (32*a^3*d*e^10*(a^3*c^7)^(1/2))/c^4 - (288*a^2*d^3*e^8*(a^3*
c^7)^(1/2))/c^3) + (d^3*e^5*(a^3*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(4*a*c) - (e^5*(a^3*c^7)^(1/2))/(4*c^7) + (5*
d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(a^3*c^7)^(1/2))/(
2*a*c^6))^(1/2)*320i)/(16*a^4*e^11 - 80*c^4*d^8*e^3 - 160*a*c^3*d^6*e^5 + 160*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e
^7 + (160*d^7*e^4*(a^3*c^7)^(1/2))/a + (160*d^5*e^6*(a^3*c^7)^(1/2))/c - (288*a*d^3*e^8*(a^3*c^7)^(1/2))/c^2 -
(32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^3) + (a*d*e^7*(a^3*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(4*a*c) - (e^5*(a^3*c^7)
^(1/2))/(4*c^7) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e
^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/(160*d^5*e^6*(a^3*c^7)^(1/2) + 16*a^4*c*e^11 - 80*c^5*d^8*e^3 - 160*
a*c^4*d^6*e^5 + 64*a^2*c^3*d^4*e^7 + 160*a^3*c^2*d^2*e^9 - (288*a*d^3*e^8*(a^3*c^7)^(1/2))/c + (160*c*d^7*e^4*
(a^3*c^7)^(1/2))/a - (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^2) + (a*c^2*d^4*e^4*(d + e*x)^(1/2)*(d^5/(4*a*c) - (e^5
*(a^3*c^7)^(1/2))/(4*c^7) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(a^3*c^7)^(1/2))/(4*a^2*c^5)
- (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 + (1
60*a^3*d^2*e^9)/c - 160*a*c*d^6*e^5 + (160*d^5*e^6*(a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(a^3*c^7)^(1/2))/c^4
- (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(a^3*c^7)^(1/2))/(a*c^2)) + (a^2*c*d^2*e^6*(d + e*x)^(1/2
)*(d^5/(4*a*c) - (e^5*(a^3*c^7)^(1/2))/(4*c^7) + (5*d^3*e^2)/(2*c^2) + (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(a^3*c^7
)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7
- 80*c^2*d^8*e^3 + (160*a^3*d^2*e^9)/c - 160*a*c*d^6*e^5 + (160*d^5*e^6*(a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*
(a^3*c^7)^(1/2))/c^4 - (32*a^2*d*e^10*(a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(a^3*c^7)^(1/2))/(a*c^2)))*(-(a^2*e^
5*(a^3*c^7)^(1/2) - a*c^6*d^5 - 5*a^3*c^4*d*e^4 - 10*a^2*c^5*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^7)^(1/2) + 10*a*c*d^
2*e^3*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*2i - (2*e*(d + e*x)^(3/2))/(3*c) - (4*d*e*(d + e*x)^(1/2))/c