3.15.49 \(\int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx\) [1449]

Optimal. Leaf size=179 \[ -\frac {2 B \sqrt {d+e x}}{c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \]

[Out]

-2*B*(e*x+d)^(1/2)/c+arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(B*a^(1/2)-A*c^(1/2))*(-e*a^(
1/2)+d*c^(1/2))^(1/2)/c^(5/4)/a^(1/2)+arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(B*a^(1/2)+A*
c^(1/2))*(e*a^(1/2)+d*c^(1/2))^(1/2)/c^(5/4)/a^(1/2)

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Rubi [A]
time = 0.42, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {839, 841, 1180, 214} \begin {gather*} \frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \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} B+A \sqrt {c}\right ) \sqrt {\sqrt {a} e+\sqrt {c} d} \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 B \sqrt {d+e x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2),x]

[Out]

(-2*B*Sqrt[d + e*x])/c + ((Sqrt[a]*B - A*Sqrt[c])*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/
Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(5/4)) + ((Sqrt[a]*B + A*Sqrt[c])*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*ArcTanh
[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[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 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*} \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx &=-\frac {2 B \sqrt {d+e x}}{c}-\frac {\int \frac {-A c d-a B e-c (B d+A e) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{c}\\ &=-\frac {2 B \sqrt {d+e x}}{c}-\frac {2 \text {Subst}\left (\int \frac {c d (B d+A e)+e (-A c d-a B e)-c (B d+A 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 B \sqrt {d+e x}}{c}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\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 )}{\sqrt {a} \sqrt {c}}+\left (-A \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )+B \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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 )\\ &=-\frac {2 B \sqrt {d+e x}}{c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \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} B+A \sqrt {c}\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} \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 = 0.35, size = 211, normalized size = 1.18 \begin {gather*} \frac {-2 \sqrt {a} B \sqrt {c} \sqrt {d+e x}-\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\left (-\sqrt {a} B+A \sqrt {c}\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a]*B*Sqrt[c]*Sqrt[d + e*x] - (Sqrt[a]*B + A*Sqrt[c])*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]*B) + A*Sqrt[c])*Sqrt[-(c*d) + S
qrt[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]*c
^(3/2))

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Maple [A]
time = 0.68, size = 190, normalized size = 1.06

method	result	size
derivativedivides	\(-\frac {2 B \sqrt {e x +d}}{c}-\frac {\left (-A c d e -B \,e^{2} a +A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A c d e +B \,e^{2} a +A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(190\)
default	\(-\frac {2 B \sqrt {e x +d}}{c}-\frac {\left (-A c d e -B \,e^{2} a +A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A c d e +B \,e^{2} a +A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(190\)
risch	\(-\frac {2 B \sqrt {e x +d}}{c}+\frac {\arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) A c d e}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) B \,e^{2} a}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) A e}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) B d}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) A c d e}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) B \,e^{2} a}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) A e}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) B d}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-2*B*(e*x+d)^(1/2)/c-(-A*c*d*e-B*e^2*a+A*(a*c*e^2)^(1/2)*e+B*(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))+(A*c*d*e+B*e^2*a+A*(a*c*e^2)^(1/2)
*e+B*(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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a),x, algorithm="maxima")

[Out]

-integrate((B*x + A)*sqrt(x*e + d)/(c*x^2 - a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1559 vs. \(2 (136) = 272\).
time = 4.72, size = 1559, normalized size = 8.71 \begin {gather*} -\frac {c \sqrt {\frac {2 \, A B a e + a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}} \log \left (-{\left (2 \, {\left (A B^{3} a c - A^{3} B c^{2}\right )} d + {\left (B^{4} a^{2} - A^{4} c^{2}\right )} e\right )} \sqrt {x e + d} + {\left (2 \, A B^{2} a c^{2} d - A a c^{4} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{3} a^{2} c + A^{2} B a c^{2}\right )} e\right )} \sqrt {\frac {2 \, A B a e + a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}}\right ) - c \sqrt {\frac {2 \, A B a e + a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}} \log \left (-{\left (2 \, {\left (A B^{3} a c - A^{3} B c^{2}\right )} d + {\left (B^{4} a^{2} - A^{4} c^{2}\right )} e\right )} \sqrt {x e + d} - {\left (2 \, A B^{2} a c^{2} d - A a c^{4} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{3} a^{2} c + A^{2} B a c^{2}\right )} e\right )} \sqrt {\frac {2 \, A B a e + a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}}\right ) + c \sqrt {\frac {2 \, A B a e - a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}} \log \left (-{\left (2 \, {\left (A B^{3} a c - A^{3} B c^{2}\right )} d + {\left (B^{4} a^{2} - A^{4} c^{2}\right )} e\right )} \sqrt {x e + d} + {\left (2 \, A B^{2} a c^{2} d + A a c^{4} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{3} a^{2} c + A^{2} B a c^{2}\right )} e\right )} \sqrt {\frac {2 \, A B a e - a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}}\right ) - c \sqrt {\frac {2 \, A B a e - a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}} \log \left (-{\left (2 \, {\left (A B^{3} a c - A^{3} B c^{2}\right )} d + {\left (B^{4} a^{2} - A^{4} c^{2}\right )} e\right )} \sqrt {x e + d} - {\left (2 \, A B^{2} a c^{2} d + A a c^{4} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{3} a^{2} c + A^{2} B a c^{2}\right )} e\right )} \sqrt {\frac {2 \, A B a e - a c^{2} \sqrt {\frac {4 \, A^{2} B^{2} c^{2} d^{2} + 4 \, {\left (A B^{3} a c + A^{3} B c^{2}\right )} d e + {\left (B^{4} a^{2} + 2 \, A^{2} B^{2} a c + A^{4} c^{2}\right )} e^{2}}{a c^{5}}} + {\left (B^{2} a + A^{2} c\right )} d}{a c^{2}}}\right ) + 4 \, \sqrt {x e + d} B}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (158) = 316\).
time = 12.58, size = 396, normalized size = 2.21 \begin {gather*} - 2 A 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 B a e^{2} \operatorname {RootSum} {\left (t^{4} \cdot \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 B d^{2} \operatorname {RootSum} {\left (t^{4} \cdot \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 B d \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 B \sqrt {d + e x}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(-c*x**2+a),x)

[Out]

-2*A*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*B*a*e**2*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(-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*B*d**2*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(-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)))) - 2*B*d*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*B*sqrt(d + e*x)/c

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (136) = 272\).
time = 1.60, size = 303, normalized size = 1.69 \begin {gather*} -\frac {2 \, \sqrt {x e + d} B}{c} - \frac {{\left (\sqrt {a c} A c^{3} d^{2} - \sqrt {a c} A a c^{2} e^{2} + {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} B {\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} A c^{3} d^{2} - \sqrt {a c} A a c^{2} e^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} B {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a),x, algorithm="giac")

[Out]

-2*sqrt(x*e + d)*B/c - (sqrt(a*c)*A*c^3*d^2 - sqrt(a*c)*A*a*c^2*e^2 + (a*c^2*d^2 - a^2*c*e^2)*B*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)*sq
rt(-c^2*d - sqrt(a*c)*c*e)) + (sqrt(a*c)*A*c^3*d^2 - sqrt(a*c)*A*a*c^2*e^2 - (a*c^2*d^2 - a^2*c*e^2)*B*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))

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Mupad [B]
time = 0.44, size = 2500, normalized size = 13.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(a - c*x^2),x)

[Out]

- 2*atanh((32*A^2*a*c^2*e^4*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) + (A^2*e*(a^3
*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) + (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16
*A^3*c^2*d^2*e^3 - 16*A^3*a*c*e^5 - 16*A*B^2*a^2*e^5 - (16*A^2*B*e^5*(a^3*c^5)^(1/2))/c^2 - (16*B^3*a*e^5*(a^3
*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 + (16*B^3*d^2*e^3*(a^3*c^5)^(1/2))/c^2 - 32*A^2*B*a*c*d*e^4 - (32*A*B^
2*d*e^4*(a^3*c^5)^(1/2))/c^2 + 16*A*B^2*a*c*d^2*e^3 + (32*A*B^2*d^3*e^2*(a^3*c^5)^(1/2))/(a*c) + (16*A^2*B*d^2
*e^3*(a^3*c^5)^(1/2))/(a*c)) + (32*B^2*a^2*c*e^4*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/
(4*a*c) + (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) + (A*B*d*(a^3*c^5)^(1/2))/(2
*a^2*c^4))^(1/2))/(16*A^3*c^2*d^2*e^3 - 16*A^3*a*c*e^5 - 16*A*B^2*a^2*e^5 - (16*A^2*B*e^5*(a^3*c^5)^(1/2))/c^2
- (16*B^3*a*e^5*(a^3*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 + (16*B^3*d^2*e^3*(a^3*c^5)^(1/2))/c^2 - 32*A^2*B
*a*c*d*e^4 - (32*A*B^2*d*e^4*(a^3*c^5)^(1/2))/c^2 + 16*A*B^2*a*c*d^2*e^3 + (32*A*B^2*d^3*e^2*(a^3*c^5)^(1/2))/
(a*c) + (16*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/(a*c)) + (32*A^2*d*e^3*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((B^2*d)/(4*
c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) + (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*
c^5) + (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*A*B^2*a^3*e^5 + 16*A^3*a^2*c*e^5 + (16*B^3*a^2*e^5*(a^3
*c^5)^(1/2))/c^3 - 16*A^3*a*c^2*d^2*e^3 + (16*A^2*B*a*e^5*(a^3*c^5)^(1/2))/c^2 + 32*A^2*B*a^2*c*d*e^4 - (32*A*
B^2*d^3*e^2*(a^3*c^5)^(1/2))/c - (16*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/c - 16*A*B^2*a^2*c*d^2*e^3 - 32*A^2*B*a*c^
2*d^3*e^2 - (16*B^3*a*d^2*e^3*(a^3*c^5)^(1/2))/c^2 + (32*A*B^2*a*d*e^4*(a^3*c^5)^(1/2))/c^2) - (32*B^2*d*e^3*(
a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) + (A^2*e*(a^3*c^5)^(1/2))/
(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) + (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*A^3*c^3*d^2*
e^3 - 16*A^3*a*c^2*e^5 - (16*A^2*B*e^5*(a^3*c^5)^(1/2))/c - 16*A*B^2*a^2*c*e^5 - (16*B^3*a*e^5*(a^3*c^5)^(1/2)
)/c^2 + 32*A^2*B*c^3*d^3*e^2 + (16*B^3*d^2*e^3*(a^3*c^5)^(1/2))/c - (32*A*B^2*d*e^4*(a^3*c^5)^(1/2))/c - 32*A^
2*B*a*c^2*d*e^4 + (32*A*B^2*d^3*e^2*(a^3*c^5)^(1/2))/a + (16*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/a + 16*A*B^2*a*c^2
*d^2*e^3) + (64*A*B*d^2*e^2*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*
c) + (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) + (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*
c^4))^(1/2))/(16*A*B^2*a^3*e^5 + 16*A^3*a^2*c*e^5 + (16*B^3*a^2*e^5*(a^3*c^5)^(1/2))/c^3 - 16*A^3*a*c^2*d^2*e^
3 + (16*A^2*B*a*e^5*(a^3*c^5)^(1/2))/c^2 + 32*A^2*B*a^2*c*d*e^4 - (32*A*B^2*d^3*e^2*(a^3*c^5)^(1/2))/c - (16*A
^2*B*d^2*e^3*(a^3*c^5)^(1/2))/c - 16*A*B^2*a^2*c*d^2*e^3 - 32*A^2*B*a*c^2*d^3*e^2 - (16*B^3*a*d^2*e^3*(a^3*c^5
)^(1/2))/c^2 + (32*A*B^2*a*d*e^4*(a^3*c^5)^(1/2))/c^2) + (64*A*B*a*c^2*d*e^3*(d + e*x)^(1/2)*((B^2*d)/(4*c^2)
+ (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) + (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5)
+ (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*A^3*c^2*d^2*e^3 - 16*A^3*a*c*e^5 - 16*A*B^2*a^2*e^5 - (16*A^
2*B*e^5*(a^3*c^5)^(1/2))/c^2 - (16*B^3*a*e^5*(a^3*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 + (16*B^3*d^2*e^3*(a^
3*c^5)^(1/2))/c^2 - 32*A^2*B*a*c*d*e^4 - (32*A*B^2*d*e^4*(a^3*c^5)^(1/2))/c^2 + 16*A*B^2*a*c*d^2*e^3 + (32*A*B
^2*d^3*e^2*(a^3*c^5)^(1/2))/(a*c) + (16*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/(a*c)))*((B^2*a*e*(a^3*c^5)^(1/2) + A^2
*c*e*(a^3*c^5)^(1/2) + A^2*a*c^4*d + B^2*a^2*c^3*d + 2*A*B*a^2*c^3*e + 2*A*B*c*d*(a^3*c^5)^(1/2))/(4*a^2*c^5))
^(1/2) - 2*atanh((32*A^2*a*c^2*e^4*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) - (A^2
*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) - (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) - (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/
2))/(16*A^3*c^2*d^2*e^3 - 16*A^3*a*c*e^5 - 16*A*B^2*a^2*e^5 + (16*A^2*B*e^5*(a^3*c^5)^(1/2))/c^2 + (16*B^3*a*e
^5*(a^3*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 - (16*B^3*d^2*e^3*(a^3*c^5)^(1/2))/c^2 - 32*A^2*B*a*c*d*e^4 + (
32*A*B^2*d*e^4*(a^3*c^5)^(1/2))/c^2 + 16*A*B^2*a*c*d^2*e^3 - (32*A*B^2*d^3*e^2*(a^3*c^5)^(1/2))/(a*c) - (16*A^
2*B*d^2*e^3*(a^3*c^5)^(1/2))/(a*c)) + (32*B^2*a^2*c*e^4*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (
A^2*d)/(4*a*c) - (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) - (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) - (A*B*d*(a^3*c^5)^(1
/2))/(2*a^2*c^4))^(1/2))/(16*A^3*c^2*d^2*e^3 - 16*A^3*a*c*e^5 - 16*A*B^2*a^2*e^5 + (16*A^2*B*e^5*(a^3*c^5)^(1/
2))/c^2 + (16*B^3*a*e^5*(a^3*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 - (16*B^3*d^2*e^3*(a^3*c^5)^(1/2))/c^2 - 3
2*A^2*B*a*c*d*e^4 + (32*A*B^2*d*e^4*(a^3*c^5)^(1/2))/c^2 + 16*A*B^2*a*c*d^2*e^3 - (32*A*B^2*d^3*e^2*(a^3*c^5)^
(1/2))/(a*c) - (16*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/(a*c)) - (32*A^2*d*e^3*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((B^2
*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) - (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) - (B^2*e*(a^3*c^5)^(1/2)
)/(4*a*c^5) - (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^...

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