3.6.32 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx\) [532]

Optimal. Leaf size=144 \[ -\frac {c d (a e-c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {a c^2 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {745, 735, 739, 212} \begin {gather*} -\frac {a c^2 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}}-\frac {c d \sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^2]/(d + e*x)^4,x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rubi steps

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

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Mathematica [A]
time = 10.15, size = 173, normalized size = 1.20 \begin {gather*} \frac {\sqrt {c d^2+a e^2} \sqrt {a+c x^2} \left (-2 a^2 e^3+c^2 d^2 x (3 d+e x)-a c e \left (5 d^2+3 d e x+2 e^2 x^2\right )\right )+3 a c^2 d (d+e x)^3 \log (d+e x)-3 a c^2 d (d+e x)^3 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{6 \left (c d^2+a e^2\right )^{5/2} (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-2*a^2*e^3 + c^2*d^2*x*(3*d + e*x) - a*c*e*(5*d^2 + 3*d*e*x + 2*e^2*x^2)
) + 3*a*c^2*d*(d + e*x)^3*Log[d + e*x] - 3*a*c^2*d*(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a +
c*x^2]])/(6*(c*d^2 + a*e^2)^(5/2)*(d + e*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(128)=256\).
time = 0.43, size = 988, normalized size = 6.86

method	result	size
default	\(\frac {-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{3 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {c d e \left (-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {c d e \left (-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}+\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{e^{2} a +c \,d^{2}}\right )}{2 e^{2} a +2 c \,d^{2}}+\frac {c \,e^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2} a +2 c \,d^{2}}\right )}{e^{2} a +c \,d^{2}}}{e^{4}}\)	\(988\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(1/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+c*d*e/(a*e^2+c*d
^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+1/2*c*d*e/(a*e^2+c
*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d*e/(a*e^2+c*d^2)*
((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-
2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-
2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))+2
*c/(a*e^2+c*d^2)*e^2*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4
*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+
c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2+c*d^2)*e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/
e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e
^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*
(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (130) = 260\).
time = 0.34, size = 418, normalized size = 2.90 \begin {gather*} -\frac {c^{3} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-7\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {5}{2}}} - \frac {\sqrt {c x^{2} + a} c^{2} d^{2}}{2 \, {\left (c^{2} d^{4} x e^{2} + c^{2} d^{5} e + 2 \, a c d^{2} x e^{4} + 2 \, a c d^{3} e^{3} + a^{2} x e^{6} + a^{2} d e^{5}\right )}} + \frac {c^{2} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-5\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d}{2 \, {\left (c^{2} d^{4} x^{2} e + c^{2} d^{6} e^{\left (-1\right )} + 2 \, c^{2} d^{5} x + 2 \, a c d^{2} x^{2} e^{3} + 4 \, a c d^{3} x e^{2} + 2 \, a c d^{4} e + a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4} + a^{2} d^{2} e^{3}\right )}} + \frac {\sqrt {c x^{2} + a} c^{2} d}{2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{3 \, {\left (c d^{2} x^{3} e^{2} + 3 \, c d^{3} x^{2} e + c d^{5} e^{\left (-1\right )} + 3 \, c d^{4} x + a x^{3} e^{4} + 3 \, a d x^{2} e^{3} + 3 \, a d^{2} x e^{2} + a d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/2*c^3*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-7)/(c*d^2*e^(-2) + a)^
(5/2) - 1/2*sqrt(c*x^2 + a)*c^2*d^2/(c^2*d^4*x*e^2 + c^2*d^5*e + 2*a*c*d^2*x*e^4 + 2*a*c*d^3*e^3 + a^2*x*e^6 +
a^2*d*e^5) + 1/2*c^2*d*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-5)/(c*d^2*e
^(-2) + a)^(3/2) - 1/2*(c*x^2 + a)^(3/2)*c*d/(c^2*d^4*x^2*e + c^2*d^6*e^(-1) + 2*c^2*d^5*x + 2*a*c*d^2*x^2*e^3
+ 4*a*c*d^3*x*e^2 + 2*a*c*d^4*e + a^2*x^2*e^5 + 2*a^2*d*x*e^4 + a^2*d^2*e^3) + 1/2*sqrt(c*x^2 + a)*c^2*d/(c^2
*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) - 1/3*(c*x^2 + a)^(3/2)/(c*d^2*x^3*e^2 + 3*c*d^3*x^2*e + c*d^5*e^(-1) + 3*c*
d^4*x + a*x^3*e^4 + 3*a*d*x^2*e^3 + 3*a*d^2*x*e^2 + a*d^3*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (130) = 260\).
time = 2.64, size = 853, normalized size = 5.92 \begin {gather*} \left [\frac {3 \, {\left (a c^{2} d x^{3} e^{3} + 3 \, a c^{2} d^{2} x^{2} e^{2} + 3 \, a c^{2} d^{3} x e + a c^{2} d^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (3 \, c^{3} d^{5} x - 3 \, a^{2} c d x e^{4} - 2 \, {\left (a^{2} c x^{2} + a^{3}\right )} e^{5} - {\left (a c^{2} d^{2} x^{2} + 7 \, a^{2} c d^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{2} - 5 \, a c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + a}}{12 \, {\left (3 \, c^{3} d^{8} x e + c^{3} d^{9} + a^{3} x^{3} e^{9} + 3 \, a^{3} d x^{2} e^{8} + 3 \, {\left (a^{2} c d^{2} x^{3} + a^{3} d^{2} x\right )} e^{7} + {\left (9 \, a^{2} c d^{3} x^{2} + a^{3} d^{3}\right )} e^{6} + 3 \, {\left (a c^{2} d^{4} x^{3} + 3 \, a^{2} c d^{4} x\right )} e^{5} + 3 \, {\left (3 \, a c^{2} d^{5} x^{2} + a^{2} c d^{5}\right )} e^{4} + {\left (c^{3} d^{6} x^{3} + 9 \, a c^{2} d^{6} x\right )} e^{3} + 3 \, {\left (c^{3} d^{7} x^{2} + a c^{2} d^{7}\right )} e^{2}\right )}}, \frac {3 \, {\left (a c^{2} d x^{3} e^{3} + 3 \, a c^{2} d^{2} x^{2} e^{2} + 3 \, a c^{2} d^{3} x e + a c^{2} d^{4}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (3 \, c^{3} d^{5} x - 3 \, a^{2} c d x e^{4} - 2 \, {\left (a^{2} c x^{2} + a^{3}\right )} e^{5} - {\left (a c^{2} d^{2} x^{2} + 7 \, a^{2} c d^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{2} - 5 \, a c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + a}}{6 \, {\left (3 \, c^{3} d^{8} x e + c^{3} d^{9} + a^{3} x^{3} e^{9} + 3 \, a^{3} d x^{2} e^{8} + 3 \, {\left (a^{2} c d^{2} x^{3} + a^{3} d^{2} x\right )} e^{7} + {\left (9 \, a^{2} c d^{3} x^{2} + a^{3} d^{3}\right )} e^{6} + 3 \, {\left (a c^{2} d^{4} x^{3} + 3 \, a^{2} c d^{4} x\right )} e^{5} + 3 \, {\left (3 \, a c^{2} d^{5} x^{2} + a^{2} c d^{5}\right )} e^{4} + {\left (c^{3} d^{6} x^{3} + 9 \, a c^{2} d^{6} x\right )} e^{3} + 3 \, {\left (c^{3} d^{7} x^{2} + a c^{2} d^{7}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/12*(3*(a*c^2*d*x^3*e^3 + 3*a*c^2*d^2*x^2*e^2 + 3*a*c^2*d^3*x*e + a*c^2*d^4)*sqrt(c*d^2 + a*e^2)*log(-(2*c^2
*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2
)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(3*c^3*d^5*x - 3*a^2*c*d*x*e^4 - 2*(a^2*c*x^2 + a^3)*e^5 - (a*c^2*d^2*x^2 + 7
*a^2*c*d^2)*e^3 + (c^3*d^4*x^2 - 5*a*c^2*d^4)*e)*sqrt(c*x^2 + a))/(3*c^3*d^8*x*e + c^3*d^9 + a^3*x^3*e^9 + 3*a
^3*d*x^2*e^8 + 3*(a^2*c*d^2*x^3 + a^3*d^2*x)*e^7 + (9*a^2*c*d^3*x^2 + a^3*d^3)*e^6 + 3*(a*c^2*d^4*x^3 + 3*a^2*
c*d^4*x)*e^5 + 3*(3*a*c^2*d^5*x^2 + a^2*c*d^5)*e^4 + (c^3*d^6*x^3 + 9*a*c^2*d^6*x)*e^3 + 3*(c^3*d^7*x^2 + a*c^
2*d^7)*e^2), 1/6*(3*(a*c^2*d*x^3*e^3 + 3*a*c^2*d^2*x^2*e^2 + 3*a*c^2*d^3*x*e + a*c^2*d^4)*sqrt(-c*d^2 - a*e^2)
*arctan(-sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(c^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) + (
3*c^3*d^5*x - 3*a^2*c*d*x*e^4 - 2*(a^2*c*x^2 + a^3)*e^5 - (a*c^2*d^2*x^2 + 7*a^2*c*d^2)*e^3 + (c^3*d^4*x^2 - 5
*a*c^2*d^4)*e)*sqrt(c*x^2 + a))/(3*c^3*d^8*x*e + c^3*d^9 + a^3*x^3*e^9 + 3*a^3*d*x^2*e^8 + 3*(a^2*c*d^2*x^3 +
a^3*d^2*x)*e^7 + (9*a^2*c*d^3*x^2 + a^3*d^3)*e^6 + 3*(a*c^2*d^4*x^3 + 3*a^2*c*d^4*x)*e^5 + 3*(3*a*c^2*d^5*x^2
+ a^2*c*d^5)*e^4 + (c^3*d^6*x^3 + 9*a*c^2*d^6*x)*e^3 + 3*(c^3*d^7*x^2 + a*c^2*d^7)*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (130) = 260\).
time = 1.33, size = 518, normalized size = 3.60 \begin {gather*} -\frac {a c^{2} d \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} d^{2} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} - a^{3} c^{\frac {5}{2}} d^{2} e^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} + 2 \, a^{4} c^{\frac {3}{2}} e^{5}}{3 \, {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-a*c^2*d*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^2*d^4 + 2*a*c*d^2*e^2
+ a^2*e^4)*sqrt(-c*d^2 - a*e^2)) + 1/3*(6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e + 4*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*c^4*d^5 - 6*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e - 14*(sqrt(c)*x - sqrt(c*x^2 + a))^3
*a*c^3*d^3*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^3 - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2
*d*e^4 + 6*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d^3*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)*d^2*
e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*e^4 + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*e^5 - a
^3*c^(5/2)*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^4 + 2*a^4*c^(3/2)*e^5)/((c^2*d^4*e^2 + 2*a*c*
d^2*e^4 + a^2*e^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(1/2)/(d + e*x)^4,x)

[Out]

int((a + c*x^2)^(1/2)/(d + e*x)^4, x)

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