\(\int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx\) [613]

Optimal result

Integrand size = 20, antiderivative size = 149 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 e \sqrt {d+e x}}{c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {718, 841, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{5/4}}-\frac {2 e \sqrt {d+e x}}{c} \]

[In]

Int[(d + e*x)^(3/2)/(a - c*x^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\frac {-2 e \sqrt {d+e x}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{c} \]

[In]

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

[Out]

(-2*e*Sqrt[d + e*x] + ((Sqrt[c]*d + Sqrt[a]*e)^2*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]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*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

Maple [A] (verified)

Time = 2.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(-2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(168\)
pseudoelliptic	\(-e \left (\frac {2 \sqrt {e x +d}}{c}+\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(168\)
default	\(2 e \left (-\frac {\sqrt {e x +d}}{c}+\frac {\left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-e^{2} a -c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(169\)
risch	\(-\frac {2 e \sqrt {e x +d}}{c}-2 e \left (\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(171\)

[In]

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

[Out]

-2*e*(1/c*(e*x+d)^(1/2)+1/2*(-e^2*a-c*d^2+2*(a*c*e^2)^(1/2)*d)/(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))-1/2*(e^2*a+c*d^2+2*(a*c*e^2)^(1/2)*d)/(a*c*e^2)^(1/
2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(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. 974 vs. \(2 (107) = 214\).

Time = 0.31 (sec) , antiderivative size = 974, normalized size of antiderivative = 6.54 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\frac {c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} - a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} - a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} + a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} + a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - 4 \, \sqrt {e x + d} e}{2 \, c} \]

[In]

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

[Out]

1/2*(c*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-
(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) + (3*a*c^2*d^2*e^2 + a^2*c*e^4 - a*c^4*d*sqrt((9*c^2*d^4
*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4
+ a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e
^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 + a^2*c*e
^4 - a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*
c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) + c*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4
*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d)
+ (3*a*c^2*d^2*e^2 + a^2*c*e^4 + a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3
+ 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt((c*d^3 + 3*a*d
*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e
^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 + a^2*c*e^4 + a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2
*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^
2))) - 4*sqrt(e*x + d)*e)/c

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (107) = 214\).

Time = 0.32 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 \, \sqrt {e x + d} e}{c} - \frac {{\left (\sqrt {a c} c^{3} d^{3} e - \sqrt {a c} a c^{2} d e^{3} + {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} c^{3} d^{3} e - \sqrt {a c} a c^{2} d e^{3} - {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.63 (sec) , antiderivative size = 1581, normalized size of antiderivative = 10.61 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(3/2)/(a - c*x^2),x)

[Out]

2*atanh((32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*
d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7 - 48*c^2*d^5*e^3 - (16*a*e^8*(a^3*c^5)^(1/2))/c^3 + 3
2*a*c*d^3*e^5 - (32*d^2*e^6*(a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(a^3*c^5)^(1/2))/(a*c)) - (32*d*e^5*(a^3*c^5)^(
1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(a^3*c^5)^(
1/2))/(4*a^2*c^4))^(1/2))/(48*c^3*d^5*e^3 - 32*a*c^2*d^3*e^5 + (16*a*e^8*(a^3*c^5)^(1/2))/c^2 - 16*a^2*c*d*e^7
- (48*d^4*e^4*(a^3*c^5)^(1/2))/a + (32*d^2*e^6*(a^3*c^5)^(1/2))/c) + (96*d^3*e^3*(a^3*c^5)^(1/2)*(d + e*x)^(1
/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4)
)^(1/2))/(16*a^3*d*e^7 - 48*a*c^2*d^5*e^3 + 32*a^2*c*d^3*e^5 - (16*a^2*e^8*(a^3*c^5)^(1/2))/c^3 + (48*d^4*e^4*
(a^3*c^5)^(1/2))/c - (32*a*d^2*e^6*(a^3*c^5)^(1/2))/c^2) + (96*a*c^2*d^2*e^4*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2
) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^
7 - 48*c^2*d^5*e^3 - (16*a*e^8*(a^3*c^5)^(1/2))/c^3 + 32*a*c*d^3*e^5 - (32*d^2*e^6*(a^3*c^5)^(1/2))/c^2 + (48*
d^4*e^4*(a^3*c^5)^(1/2))/(a*c)))*((a*c^4*d^3 - a*e^3*(a^3*c^5)^(1/2) + 3*a^2*c^3*d*e^2 - 3*c*d^2*e*(a^3*c^5)^(
1/2))/(4*a^2*c^5))^(1/2) - 2*atanh((32*d*e^5*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c)
+ (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(32*a*c^2*d^3*e^5 - 48*c^3*d
^5*e^3 + (16*a*e^8*(a^3*c^5)^(1/2))/c^2 + 16*a^2*c*d*e^7 - (48*d^4*e^4*(a^3*c^5)^(1/2))/a + (32*d^2*e^6*(a^3*c
^5)^(1/2))/c) - (32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) + (e^3*(a^3*c^5)^(1/2))/(4*a*c^
5) + (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7 - 48*c^2*d^5*e^3 + (16*a*e^8*(a^3*c^5)^(1/2))
/c^3 + 32*a*c*d^3*e^5 + (32*d^2*e^6*(a^3*c^5)^(1/2))/c^2 - (48*d^4*e^4*(a^3*c^5)^(1/2))/(a*c)) + (96*d^3*e^3*(
a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) + (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(
a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^3*d*e^7 - 48*a*c^2*d^5*e^3 + 32*a^2*c*d^3*e^5 + (16*a^2*e^8*(a^3*c^5
)^(1/2))/c^3 - (48*d^4*e^4*(a^3*c^5)^(1/2))/c + (32*a*d^2*e^6*(a^3*c^5)^(1/2))/c^2) - (96*a*c^2*d^2*e^4*(d + e
*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) + (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^
2*c^4))^(1/2))/(16*a^2*d*e^7 - 48*c^2*d^5*e^3 + (16*a*e^8*(a^3*c^5)^(1/2))/c^3 + 32*a*c*d^3*e^5 + (32*d^2*e^6*
(a^3*c^5)^(1/2))/c^2 - (48*d^4*e^4*(a^3*c^5)^(1/2))/(a*c)))*((a*c^4*d^3 + a*e^3*(a^3*c^5)^(1/2) + 3*a^2*c^3*d*
e^2 + 3*c*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^5))^(1/2) - (2*e*(d + e*x)^(1/2))/c