3.5.52 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx\)

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

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Rubi [A]  time = 0.09, 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 = {731, 721, 725, 206} \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]

-(c*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(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 206

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

Rule 721

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 725

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 731

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 ) \operatorname {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 = 0.17, size = 173, normalized size = 1.20 \begin {gather*} \frac {\sqrt {a+c x^2} \sqrt {a e^2+c d^2} \left (-2 a^2 e^3-a c e \left (5 d^2+3 d e x+2 e^2 x^2\right )+c^2 d^2 x (3 d+e x)\right )-3 a c^2 d (d+e x)^3 \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )+3 a c^2 d (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left (a e^2+c d^2\right )^{5/2}} \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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IntegrateAlgebraic [B]  time = 35.34, size = 2035, normalized size = 14.13 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(-32*a^4*d^4*e^5 - 88*a^4*d^3*e^6*x - 120*a^4*d^2*e^7*x^2 - 88*a^4*d*e^8*x^3 - 26*a^4*e^9*x^4
) + Sqrt[c]*(96*a^4*d^4*e^5*x + 264*a^4*d^3*e^6*x^2 + 360*a^4*d^2*e^7*x^3 + 264*a^4*d*e^8*x^4 + 78*a^4*e^9*x^5
) + c*Sqrt[a + c*x^2]*(-80*a^3*d^6*e^3 - 256*a^3*d^5*e^4*x - 552*a^3*d^4*e^5*x^2 - 760*a^3*d^3*e^6*x^3 - 725*a
^3*d^2*e^7*x^4 - 459*a^3*d*e^8*x^5 - 132*a^3*e^9*x^6) + c^(3/2)*(-32*a^3*d^7*e^2 + 80*a^3*d^6*e^3*x + 432*a^3*
d^5*e^4*x^2 + 1016*a^3*d^4*e^5*x^3 + 1318*a^3*d^3*e^6*x^4 + 1113*a^3*d^2*e^7*x^5 + 651*a^3*d*e^8*x^6 + 186*a^3
*e^9*x^7) + c^4*Sqrt[a + c*x^2]*(-64*d^9*x^3 - 320*d^8*e*x^4 - 576*d^7*e^2*x^5 - 608*d^6*e^3*x^6 - 388*d^5*e^4
*x^7 - 108*d^4*e^5*x^8) + c^2*Sqrt[a + c*x^2]*(120*a^2*d^7*e^2*x + 120*a^2*d^6*e^3*x^2 - 240*a^2*d^5*e^4*x^3 -
912*a^2*d^4*e^5*x^4 - 1167*a^2*d^3*e^6*x^5 - 798*a^2*d^2*e^7*x^6 - 381*a^2*d*e^8*x^7 - 108*a^2*e^9*x^8) + c^(
9/2)*(64*d^9*x^4 + 320*d^8*e*x^5 + 576*d^7*e^2*x^6 + 608*d^6*e^3*x^7 + 388*d^5*e^4*x^8 + 108*d^4*e^5*x^9) + c^
3*Sqrt[a + c*x^2]*(-48*a*d^9*x - 240*a*d^8*e*x^2 - 256*a*d^7*e^2*x^3 - 56*a*d^6*e^3*x^4 + 93*a*d^5*e^4*x^5 - 6
5*a*d^4*e^5*x^6 - 226*a*d^3*e^6*x^7 - 90*a*d^2*e^7*x^8 + 12*a*d*e^8*x^9) + c^(5/2)*(16*a^2*d^9 + 80*a^2*d^8*e*
x - 64*a^2*d^7*e^2*x^2 - 168*a^2*d^6*e^3*x^3 + 145*a^2*d^5*e^4*x^4 + 931*a^2*d^4*e^5*x^5 + 1280*a^2*d^3*e^6*x^
6 + 843*a^2*d^2*e^7*x^7 + 375*a^2*d*e^8*x^8 + 108*a^2*e^9*x^9) + c^(7/2)*(80*a*d^9*x^2 + 400*a*d^8*e*x^3 + 544
*a*d^7*e^2*x^4 + 360*a*d^6*e^3*x^5 + 101*a*d^5*e^4*x^6 + 119*a*d^4*e^5*x^7 + 226*a*d^3*e^6*x^8 + 90*a*d^2*e^7*
x^9 - 12*a*d*e^8*x^10))/(6*a^4*e^6*(d + e*x)^3*(2*d + e*x)^4 + 24*c^4*d^4*e^2*x^4*(d + e*x)^3*(2*d + e*x)^4 -
18*a^3*Sqrt[c]*e^6*x*(d + e*x)^3*(2*d + e*x)^4*Sqrt[a + c*x^2] - 24*c^(7/2)*d^4*e^2*x^3*(d + e*x)^3*(2*d + e*x
)^4*Sqrt[a + c*x^2] + 6*c*e^2*(d + e*x)^3*(2*d + e*x)^4*(2*a^3*d^2*e^2 + 5*a^3*e^4*x^2) + 6*c^(5/2)*e^2*(d + e
*x)^3*(2*d + e*x)^4*Sqrt[a + c*x^2]*(-3*a*d^4*x - 8*a*d^2*e^2*x^3) + 6*c^(3/2)*e^2*(d + e*x)^3*(2*d + e*x)^4*S
qrt[a + c*x^2]*(-6*a^2*d^2*e^2*x - 4*a^2*e^4*x^3) + 6*c^3*e^2*(d + e*x)^3*(2*d + e*x)^4*(5*a*d^4*x^2 + 8*a*d^2
*e^2*x^4) + 6*c^2*e^2*(d + e*x)^3*(2*d + e*x)^4*(a^2*d^4 + 10*a^2*d^2*e^2*x^2 + 4*a^2*e^4*x^4)) + (4*x*Sqrt[a
+ c*x^2] + (16*a*d*ArcTan[(-(Sqrt[c]*d) - Sqrt[c]*e*x + e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/Sqrt[-(c*d
^2) - a*e^2] + (32*a*e*x*ArcTan[(-(Sqrt[c]*d) - Sqrt[c]*e*x + e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/Sqrt
[-(c*d^2) - a*e^2] + (24*a*e^2*x^2*ArcTan[(-(Sqrt[c]*d) - Sqrt[c]*e*x + e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e
^2]])/(d*Sqrt[-(c*d^2) - a*e^2]) + (8*a*e^3*x^3*ArcTan[(-(Sqrt[c]*d) - Sqrt[c]*e*x + e*Sqrt[a + c*x^2])/Sqrt[-
(c*d^2) - a*e^2]])/(d^2*Sqrt[-(c*d^2) - a*e^2]) + (a*e^4*x^4*ArcTan[(-(Sqrt[c]*d) - Sqrt[c]*e*x + e*Sqrt[a + c
*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(d^3*Sqrt[-(c*d^2) - a*e^2]))/(2*d + e*x)^4 - (a^3*e^4*ArcTan[(Sqrt[c]*d)/Sqrt
[-(c*d^2) - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) - a*e^2] - (e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(d^3*
Sqrt[-(c*d^2) - a*e^2]*(c*d^2 + a*e^2)^2) + (2*a^2*e^2*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) - a*e^2] + (Sqrt[c]*e*
x)/Sqrt[-(c*d^2) - a*e^2] - (e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(d^3*Sqrt[-(c*d^2) - a*e^2]*(c*d^2 +
a*e^2))

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fricas [B]  time = 0.83, size = 861, normalized size = 5.98 \begin {gather*} \left [\frac {3 \, {\left (a c^{2} d e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{2} x^{2} + 3 \, a c^{2} d^{3} e x + a c^{2} d^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (5 \, a c^{2} d^{4} e + 7 \, a^{2} c d^{2} e^{3} + 2 \, a^{3} e^{5} - {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3} - 2 \, a^{2} c e^{5}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} - a^{2} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{9} + 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} + a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} + 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} + 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} + a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e + 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} + a^{3} d^{2} e^{7}\right )} x\right )}}, -\frac {3 \, {\left (a c^{2} d e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{2} x^{2} + 3 \, a c^{2} d^{3} e x + 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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (5 \, a c^{2} d^{4} e + 7 \, a^{2} c d^{2} e^{3} + 2 \, a^{3} e^{5} - {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3} - 2 \, a^{2} c e^{5}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} - a^{2} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{9} + 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} + a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} + 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} + 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} + a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e + 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} + a^{3} d^{2} e^{7}\right )} x\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*e^3*x^3 + 3*a*c^2*d^2*e^2*x^2 + 3*a*c^2*d^3*e*x + a*c^2*d^4)*sqrt(c*d^2 + a*e^2)*log((2*a*c*
d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))
/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(5*a*c^2*d^4*e + 7*a^2*c*d^2*e^3 + 2*a^3*e^5 - (c^3*d^4*e - a*c^2*d^2*e^3 - 2*
a^2*c*e^5)*x^2 - 3*(c^3*d^5 - a^2*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^9 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 +
a^3*d^3*e^6 + (c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 + 3*a*c^2*d^5*e
^4 + 3*a^2*c*d^3*e^6 + a^3*d*e^8)*x^2 + 3*(c^3*d^8*e + 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 + a^3*d^2*e^7)*x), -1
/6*(3*(a*c^2*d*e^3*x^3 + 3*a*c^2*d^2*e^2*x^2 + 3*a*c^2*d^3*e*x + 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)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + (5*a*c^2*d^4*e +
7*a^2*c*d^2*e^3 + 2*a^3*e^5 - (c^3*d^4*e - a*c^2*d^2*e^3 - 2*a^2*c*e^5)*x^2 - 3*(c^3*d^5 - a^2*c*d*e^4)*x)*sq
rt(c*x^2 + a))/(c^3*d^9 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 + a^3*d^3*e^6 + (c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3
*a^2*c*d^2*e^7 + a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 + 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 + a^3*d*e^8)*x^2 + 3*(c^3*d
^8*e + 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 + a^3*d^2*e^7)*x)]

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giac [B]  time = 0.26, 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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maple [B]  time = 0.06, size = 1262, normalized size = 8.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/e^2/(a*e^2+c*d^2)/(x+d/e)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-1/2/e*c*d/(a*e^2+c*d^2
)^2/(x+d/e)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-1/2*c^2*d^2/(a*e^2+c*d^2)^3/(x+d/e)*(-2*(
x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-1/2/e*c^3*d^3/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+
(a*e^2+c*d^2)/e^2)^(1/2)+1/2/e^2*c^(7/2)*d^4/(a*e^2+c*d^2)^3*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2*(x+d/e)*c*d/e+(
x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))+1/2/e*c^3*d^3/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*
d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x
+d/e))*a+1/2/e^3*c^4*d^5/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*
((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+1/2*c^3*d^2/(a*e^2+
c*d^2)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x+1/2*c^(5/2)*d^2/(a*e^2+c*d^2)^3*ln((-c*d/e+(
x+d/e)*c)/c^(1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))*a+1/2/e*c^2*d/(a*e^2+c*d^2)^2*(-2*(x
+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-1/2/e^2*c^(5/2)*d^2/(a*e^2+c*d^2)^2*ln((-c*d/e+(x+d/e)*c)/c^(
1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))-1/2/e*c^2*d/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(
1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2
+c*d^2)/e^2)^(1/2))/(x+d/e))*a-1/2/e^3*c^3*d^3/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+
2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e
))

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maxima [B]  time = 1.76, size = 428, normalized size = 2.97 \begin {gather*} -\frac {\sqrt {c x^{2} + a} c^{2} d^{2}}{2 \, {\left (c^{2} d^{4} e^{2} x + 2 \, a c d^{2} e^{4} x + a^{2} e^{6} x + c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d}{2 \, {\left (c^{2} d^{4} e x^{2} + 2 \, a c d^{2} e^{3} x^{2} + a^{2} e^{5} x^{2} + 2 \, c^{2} d^{5} x + 4 \, a c d^{3} e^{2} x + 2 \, a^{2} d e^{4} x + \frac {c^{2} d^{6}}{e} + 2 \, a c d^{4} e + 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} e^{2} x^{3} + a e^{4} x^{3} + 3 \, c d^{3} e x^{2} + 3 \, a d e^{3} x^{2} + 3 \, c d^{4} x + 3 \, a d^{2} e^{2} x + \frac {c d^{5}}{e} + a d^{3} e\right )}} - \frac {c^{3} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{2}} e^{7}} + \frac {c^{2} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{5}} \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*sqrt(c*x^2 + a)*c^2*d^2/(c^2*d^4*e^2*x + 2*a*c*d^2*e^4*x + a^2*e^6*x + c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*
e^5) - 1/2*(c*x^2 + a)^(3/2)*c*d/(c^2*d^4*e*x^2 + 2*a*c*d^2*e^3*x^2 + a^2*e^5*x^2 + 2*c^2*d^5*x + 4*a*c*d^3*e^
2*x + 2*a^2*d*e^4*x + c^2*d^6/e + 2*a*c*d^4*e + 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*e^2*x^3 + a*e^4*x^3 + 3*c*d^3*e*x^2 + 3*a*d*e^3*x^2 + 3*c*d^4*
x + 3*a*d^2*e^2*x + c*d^5/e + a*d^3*e) - 1/2*c^3*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*a
bs(e*x + d)))/((a + c*d^2/e^2)^(5/2)*e^7) + 1/2*c^2*d*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*
abs(e*x + d)))/((a + c*d^2/e^2)^(3/2)*e^5)

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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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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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