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

Optimal. Leaf size=149 \[ -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{5/4}}-\frac {2 e \sqrt {d+e x}}{c} \]

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Rubi [A]  time = 0.28, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {704, 827, 1166, 208} \begin {gather*} -\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 {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 704

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 827

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 1166

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*} \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx &=-\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 \operatorname {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 \operatorname {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 \operatorname {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*}

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Mathematica [A]  time = 0.11, size = 147, normalized size = 0.99 \begin {gather*} \frac {-2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x}+\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (-\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )\right )+\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.39, size = 223, normalized size = 1.50 \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{\sqrt {a} c \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {2 e \sqrt {d+e x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*Sqrt[d + e*x])/c + ((Sqrt[c]*d + Sqrt[a]*e)^2*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(S
qrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*c*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*ArcT
an[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*c*Sqrt[-(Sqrt[c]*(Sqrt[
c]*d - Sqrt[a]*e))])

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fricas [B]  time = 0.42, size = 974, normalized size = 6.54 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.30, size = 304, normalized size = 2.04 \begin {gather*} -\frac {{\left (\sqrt {a c} c^{3} d^{3} - \sqrt {a c} a c^{2} d e^{2} + {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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}} + \frac {{\left (\sqrt {a c} c^{3} d^{3} - \sqrt {a c} a c^{2} d e^{2} - {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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}} - \frac {2 \, \sqrt {x e + d} e}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 335, normalized size = 2.25 \begin {gather*} \frac {a \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\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 {a \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\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 {c \,d^{2} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\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 {c \,d^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\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 {2 d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 \sqrt {e x +d}\, e}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(e*x+d)^(1/2)/c*e+1/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(
1/2))*c)^(1/2)*c)*a*e^3+e/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2
)^(1/2))*c)^(1/2)*c)*c*d^2+2*e/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)
^(1/2)*c)*d+1/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)
^(1/2)*c)*a*e^3+e/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2)
)*c)^(1/2)*c)*c*d^2-2*e/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)
*c)*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} - a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.68, size = 1581, normalized size = 10.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 65.63, size = 394, normalized size = 2.64 \begin {gather*} - \frac {2 a e^{3} \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left (t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )}}{c} + 2 d^{2} e \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left (t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )} - 2 d e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left (t \mapsto t \log {\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} - 2 d e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left (t \mapsto t \log {\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} - \frac {2 e \sqrt {d + e x}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*e**3*RootSum(_t**4*(256*a**3*c*e**6 - 256*a**2*c**2*d**2*e**4) + 32*_t**2*a*c*d*e**2 - 1, Lambda(_t, _t*l
og(-64*_t**3*a**2*c*d*e**4 + 64*_t**3*a*c**2*d**3*e**2 - 4*_t*a*e**2 - 4*_t*c*d**2 + sqrt(d + e*x))))/c + 2*d*
*2*e*RootSum(_t**4*(256*a**3*c*e**6 - 256*a**2*c**2*d**2*e**4) + 32*_t**2*a*c*d*e**2 - 1, Lambda(_t, _t*log(-6
4*_t**3*a**2*c*d*e**4 + 64*_t**3*a*c**2*d**3*e**2 - 4*_t*a*e**2 - 4*_t*c*d**2 + sqrt(d + e*x)))) - 2*d*e*RootS
um(256*_t**4*a**2*c**3*e**4 - 32*_t**2*a*c**2*d*e**2 - a*e**2 + c*d**2, Lambda(_t, _t*log(-64*_t**3*a*c**2*e**
2 + 4*_t*c*d + sqrt(d + e*x)))) - 2*d*e*RootSum(256*_t**4*a**2*c**3*e**4 - 32*_t**2*a*c**2*d*e**2 - a*e**2 + c
*d**2, Lambda(_t, _t*log(-64*_t**3*a*c**2*e**2 + 4*_t*c*d + sqrt(d + e*x)))) - 2*e*sqrt(d + e*x)/c

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