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

Optimal. Leaf size=167 \[ -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{7/4}}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {4 d e \sqrt {d+e x}}{c} \]

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Rubi [A]  time = 0.40, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {704, 825, 827, 1166, 208} \begin {gather*} -\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 {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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 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 825

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 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)^{5/2}}{a-c x^2} \, dx &=-\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 \operatorname {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 \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} c}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^3 \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} 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*}

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Mathematica [A]  time = 0.26, size = 158, normalized size = 0.95 \begin {gather*} \frac {-2 \sqrt {a} c^{3/4} e \sqrt {d+e x} (7 d+e x)-3 \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 )+3 \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{3 \sqrt {a} c^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.52, size = 242, normalized size = 1.45 \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^3 \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^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^3 \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^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {2 e \left ((d+e x)^{3/2}+6 d \sqrt {d+e x}\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*(6*d*Sqrt[d + e*x] + (d + e*x)^(3/2)))/(3*c) + ((Sqrt[c]*d + Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*
Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*c^(3/2)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))])
- ((Sqrt[c]*d - Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)]
)/(Sqrt[a]*c^(3/2)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 0.52, size = 1617, normalized size = 9.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.33, size = 402, normalized size = 2.41 \begin {gather*} -\frac {{\left (\sqrt {a c} c^{4} d^{4} + 3 \, \sqrt {a c} a c^{3} d^{2} e^{2} - {\left (3 \, \sqrt {a c} a c d^{2} e^{2} + \sqrt {a c} a^{2} e^{4}\right )} c^{2} + 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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}} + \frac {{\left (\sqrt {a c} c^{4} d^{4} + 3 \, \sqrt {a c} a c^{3} d^{2} e^{2} - {\left (3 \, \sqrt {a c} a c d^{2} e^{2} + \sqrt {a c} a^{2} e^{4}\right )} c^{2} - 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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}} - \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {x e + d} c^{2} d e\right )}}{3 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(a*c)*c^4*d^4 + 3*sqrt(a*c)*a*c^3*d^2*e^2 - (3*sqrt(a*c)*a*c*d^2*e^2 + sqrt(a*c)*a^2*e^4)*c^2 + 2*(a*c^3
*d^3*e - a^2*c^2*d*e^3)*abs(c))*arctan(sqrt(x*e + 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)) + (sqrt(a*c)*c^4*d^4 + 3*sqrt(a*c)*a*c^3*d
^2*e^2 - (3*sqrt(a*c)*a*c*d^2*e^2 + sqrt(a*c)*a^2*e^4)*c^2 - 2*(a*c^3*d^3*e - a^2*c^2*d*e^3)*abs(c))*arctan(sq
rt(x*e + 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)*sqr
t(-c^2*d + sqrt(a*c)*c*e)) - 2/3*((x*e + d)^(3/2)*c^2*e + 6*sqrt(x*e + d)*c^2*d*e)/c^3

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maple [B]  time = 0.15, size = 460, normalized size = 2.75 \begin {gather*} \frac {3 a d \,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 {3 a d \,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^{3} 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^{3} 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 {a \,e^{3} \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}\, 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 {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c}+\frac {3 d^{2} 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 {3 d^{2} 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 {4 \sqrt {e x +d}\, d e}{c}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} e}{3 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 {5}{2}}}{c x^{2} - a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.81, size = 3385, normalized size = 20.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 124.49, size = 498, normalized size = 2.98 \begin {gather*} - \frac {4 a d 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} - \frac {2 a e^{3} \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 )}}{c} + 4 d^{3} 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 )} - 4 d^{2} 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^{2} 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 {4 d e \sqrt {d + e x}}{c} - \frac {2 e \left (d + e x\right )^{\frac {3}{2}}}{3 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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