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

Optimal. Leaf size=111 \[ \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {658, 650} \begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d^2 - a*e^2)*(d + e*x)^4) + (4*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^3)

Rule 650

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

Rule 658

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {(2 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 61, normalized size = 0.55 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (c d (5 d+2 e x)-3 a e^2\right )}{15 (d+e x)^4 \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-3*a*e^2 + c*d*(5*d + 2*e*x)))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^4)

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IntegrateAlgebraic [B]  time = 8.94, size = 3233, normalized size = 29.13 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c*d*e]*Sqrt[a*d*e + c*d^2*x + a*e^2*x + c*d*e*x^2]*(5*a*c^5*d^15*e - 18*a^2*c^4*d^13*e^3 + 24*a^3*c^3*
d^11*e^5 - 14*a^4*c^2*d^9*e^7 + 3*a^5*c*d^7*e^9 + 5*c^6*d^16*x - 56*a*c^5*d^14*e^2*x + 112*a^2*c^4*d^12*e^4*x
+ 10*a^3*c^3*d^10*e^6*x - 217*a^4*c^2*d^8*e^8*x + 206*a^5*c*d^6*e^10*x - 60*a^6*d^4*e^12*x - 38*c^6*d^15*e*x^2
+ 177*a*c^5*d^13*e^3*x^2 + 94*a^2*c^4*d^11*e^5*x^2 - 724*a^3*c^3*d^9*e^7*x^2 + 408*a^4*c^2*d^7*e^9*x^2 + 323*
a^5*c*d^5*e^11*x^2 - 240*a^6*d^3*e^13*x^2 + 89*c^6*d^14*e^2*x^3 + 342*a*c^5*d^12*e^4*x^3 - 1920*a^2*c^4*d^10*e
^6*x^3 + 538*a^3*c^3*d^8*e^8*x^3 + 2511*a^4*c^2*d^6*e^10*x^3 - 1200*a^5*c*d^4*e^12*x^3 - 360*a^6*d^2*e^14*x^3
+ 192*c^6*d^13*e^3*x^4 - 2656*a*c^5*d^11*e^5*x^4 + 1856*a^2*c^4*d^9*e^7*x^4 + 4768*a^3*c^3*d^7*e^9*x^4 - 400*a
^4*c^2*d^5*e^11*x^4 - 3520*a^5*c*d^3*e^13*x^4 - 240*a^6*d*e^15*x^4 - 1140*c^6*d^12*e^4*x^5 + 2280*a*c^5*d^10*e
^6*x^5 + 6476*a^2*c^4*d^8*e^8*x^5 + 1232*a^3*c^3*d^6*e^10*x^5 - 9852*a^4*c^2*d^4*e^12*x^5 - 3160*a^5*c*d^2*e^1
4*x^5 - 60*a^6*e^16*x^5 + 960*c^6*d^11*e^5*x^6 + 5952*a*c^5*d^9*e^7*x^6 - 2304*a^2*c^4*d^7*e^9*x^6 - 12800*a^3
*c^3*d^5*e^11*x^6 - 11584*a^4*c^2*d^3*e^13*x^6 - 960*a^5*c*d*e^15*x^6 + 1856*c^6*d^10*e^6*x^7 - 4864*a*c^5*d^8
*e^8*x^7 - 11008*a^2*c^4*d^6*e^10*x^7 - 16896*a^3*c^3*d^4*e^12*x^7 - 4032*a^4*c^2*d^2*e^14*x^7 - 2048*c^6*d^9*
e^7*x^8 - 6144*a*c^5*d^7*e^9*x^8 - 10240*a^2*c^4*d^5*e^11*x^8 - 6144*a^3*c^3*d^3*e^13*x^8 - 1024*c^6*d^8*e^8*x
^9 - 2048*a*c^5*d^6*e^10*x^9 - 3072*a^2*c^4*d^4*e^12*x^9) + 2*(8*a^3*c^4*d^13*e^5 - 30*a^4*c^3*d^11*e^7 + 42*a
^5*c^2*d^9*e^9 - 26*a^6*c*d^7*e^11 + 6*a^7*d^5*e^13 + 50*a^2*c^5*d^14*e^4*x - 170*a^3*c^4*d^12*e^6*x + 180*a^4
*c^3*d^10*e^8*x - 20*a^5*c^2*d^8*e^10*x - 70*a^6*c*d^6*e^12*x + 30*a^7*d^4*e^14*x + 100*a*c^6*d^15*e^3*x^2 - 5
40*a^2*c^5*d^13*e^5*x^2 + 560*a^3*c^4*d^11*e^7*x^2 + 500*a^4*c^3*d^9*e^9*x^2 - 960*a^5*c^2*d^7*e^11*x^2 + 280*
a^6*c*d^5*e^13*x^2 + 60*a^7*d^3*e^15*x^2 + 50*c^7*d^16*e^2*x^3 - 690*a*c^6*d^14*e^4*x^3 + 1120*a^2*c^5*d^12*e^
6*x^3 + 1440*a^3*c^4*d^10*e^8*x^3 - 2310*a^4*c^3*d^8*e^10*x^3 - 970*a^5*c^2*d^6*e^12*x^3 + 1300*a^6*c*d^4*e^14
*x^3 + 60*a^7*d^2*e^16*x^3 - 280*c^7*d^15*e^3*x^4 + 1000*a*c^6*d^13*e^5*x^4 + 2800*a^2*c^5*d^11*e^7*x^4 - 4400
*a^3*c^4*d^9*e^9*x^4 - 5510*a^4*c^3*d^7*e^11*x^4 + 4450*a^5*c^2*d^5*e^13*x^4 + 1910*a^6*c*d^3*e^15*x^4 + 30*a^
7*d*e^17*x^4 + 330*c^7*d^14*e^4*x^5 + 2870*a*c^6*d^12*e^6*x^5 - 7530*a^2*c^5*d^10*e^8*x^5 - 8798*a^3*c^4*d^8*e
^10*x^5 + 5390*a^4*c^3*d^6*e^12*x^5 + 10722*a^5*c^2*d^4*e^14*x^5 + 1234*a^6*c*d^2*e^16*x^5 + 6*a^7*e^18*x^5 +
980*c^7*d^13*e^5*x^6 - 6600*a*c^6*d^11*e^7*x^6 - 8140*a^2*c^5*d^9*e^9*x^6 + 6000*a^3*c^4*d^7*e^11*x^6 + 23820*
a^4*c^3*d^5*e^13*x^6 + 8600*a^5*c^2*d^3*e^15*x^6 + 300*a^6*c*d*e^17*x^6 - 2080*c^7*d^12*e^6*x^7 - 4640*a*c^6*d
^10*e^8*x^7 + 10240*a^2*c^5*d^8*e^10*x^7 + 27520*a^3*c^4*d^6*e^12*x^7 + 22240*a^4*c^3*d^4*e^14*x^7 + 2400*a^5*
c^2*d^2*e^16*x^7 - 960*c^7*d^11*e^7*x^8 + 8960*a*c^6*d^9*e^9*x^8 + 19200*a^2*c^5*d^7*e^11*x^8 + 25600*a^3*c^4*
d^5*e^13*x^8 + 6720*a^4*c^3*d^3*e^15*x^8 + 2560*c^7*d^10*e^8*x^9 + 7680*a*c^6*d^8*e^10*x^9 + 12800*a^2*c^5*d^6
*e^12*x^9 + 7680*a^3*c^4*d^4*e^14*x^9 + 1024*c^7*d^9*e^9*x^10 + 2048*a*c^6*d^7*e^11*x^10 + 3072*a^2*c^5*d^5*e^
13*x^10))/(15*c*d^5*Sqrt[a*d*e + c*d^2*x + a*e^2*x + c*d*e*x^2]*(10*c^5*d^13*e^2*x - 40*a*c^4*d^11*e^4*x + 60*
a^2*c^3*d^9*e^6*x - 40*a^3*c^2*d^7*e^8*x + 10*a^4*c*d^5*e^10*x - 40*c^5*d^12*e^3*x^2 + 80*a*c^4*d^10*e^5*x^2 -
80*a^3*c^2*d^6*e^9*x^2 + 40*a^4*c*d^4*e^11*x^2 - 20*c^5*d^11*e^4*x^3 + 320*a*c^4*d^9*e^6*x^3 - 520*a^2*c^3*d^
7*e^8*x^3 + 160*a^3*c^2*d^5*e^10*x^3 + 60*a^4*c*d^3*e^12*x^3 + 280*c^5*d^10*e^5*x^4 - 560*a*c^4*d^8*e^7*x^4 -
320*a^2*c^3*d^6*e^9*x^4 + 560*a^3*c^2*d^4*e^11*x^4 + 40*a^4*c*d^2*e^13*x^4 - 70*c^5*d^9*e^6*x^5 - 1240*a*c^4*d
^7*e^8*x^5 + 1292*a^2*c^3*d^5*e^10*x^5 + 520*a^3*c^2*d^3*e^12*x^5 + 10*a^4*c*d*e^14*x^5 - 800*c^5*d^8*e^7*x^6
+ 864*a*c^4*d^6*e^9*x^6 + 1824*a^2*c^3*d^4*e^11*x^6 + 160*a^3*c^2*d^2*e^13*x^6 + 32*c^5*d^7*e^8*x^7 + 2368*a*c
^4*d^5*e^10*x^7 + 672*a^2*c^3*d^3*e^12*x^7 + 1024*c^5*d^6*e^9*x^8 + 1024*a*c^4*d^4*e^11*x^8 + 512*c^5*d^5*e^10
*x^9) + 15*c*d^5*Sqrt[c*d*e]*(c^5*d^15 - 5*a*c^4*d^13*e^2 + 10*a^2*c^3*d^11*e^4 - 10*a^3*c^2*d^9*e^6 + 5*a^4*c
*d^7*e^8 - a^5*d^5*e^10 - 5*c^5*d^14*e*x + 15*a*c^4*d^12*e^3*x - 10*a^2*c^3*d^10*e^5*x - 10*a^3*c^2*d^8*e^7*x
+ 15*a^4*c*d^6*e^9*x - 5*a^5*d^4*e^11*x + 70*a*c^4*d^11*e^4*x^2 - 200*a^2*c^3*d^9*e^6*x^2 + 180*a^3*c^2*d^7*e^
8*x^2 - 40*a^4*c*d^5*e^10*x^2 - 10*a^5*d^3*e^12*x^2 + 70*c^5*d^12*e^3*x^3 - 290*a*c^4*d^10*e^5*x^3 + 140*a^2*c
^3*d^8*e^7*x^3 + 300*a^3*c^2*d^6*e^9*x^3 - 210*a^4*c*d^4*e^11*x^3 - 10*a^5*d^2*e^13*x^3 - 95*c^5*d^11*e^4*x^4
- 345*a*c^4*d^9*e^6*x^4 + 1290*a^2*c^3*d^7*e^8*x^4 - 530*a^3*c^2*d^5*e^10*x^4 - 315*a^4*c*d^3*e^12*x^4 - 5*a^5
*d*e^14*x^4 - 321*c^5*d^10*e^5*x^5 + 1395*a*c^4*d^8*e^7*x^5 + 270*a^2*c^3*d^6*e^9*x^5 - 1650*a^3*c^2*d^4*e^11*
x^5 - 205*a^4*c*d^2*e^13*x^5 - a^5*e^15*x^5 + 430*c^5*d^9*e^6*x^6 + 1480*a*c^4*d^7*e^8*x^6 - 3020*a^2*c^3*d^5*
e^10*x^6 - 1400*a^3*c^2*d^3*e^12*x^6 - 50*a^4*c*d*e^14*x^6 + 880*c^5*d^8*e^7*x^7 - 2160*a*c^4*d^6*e^9*x^7 - 34
40*a^2*c^3*d^4*e^11*x^7 - 400*a^3*c^2*d^2*e^13*x^7 - 480*c^5*d^7*e^8*x^8 - 3520*a*c^4*d^5*e^10*x^8 - 1120*a^2*
c^3*d^3*e^12*x^8 - 1280*c^5*d^6*e^9*x^9 - 1280*a*c^4*d^4*e^11*x^9 - 512*c^5*d^5*e^10*x^10))

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fricas [A]  time = 0.96, size = 205, normalized size = 1.85 \begin {gather*} \frac {2 \, {\left (2 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 3 \, a^{2} e^{3} + {\left (5 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (c^{2} d^{7} - 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} + {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \, {\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \, {\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*c^2*d^2*e*x^2 + 5*a*c*d^2*e - 3*a^2*e^3 + (5*c^2*d^3 - a*c*d*e^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)/(c^2*d^7 - 2*a*c*d^5*e^2 + a^2*d^3*e^4 + (c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*x^3 + 3*(c^2*d^5*e
^2 - 2*a*c*d^3*e^4 + a^2*d*e^6)*x^2 + 3*(c^2*d^6*e - 2*a*c*d^4*e^3 + a^2*d^2*e^5)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 90, normalized size = 0.81 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-2 c d e x +3 a \,e^{2}-5 c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (e x +d \right )^{3} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(c*d*x+a*e)*(-2*c*d*e*x+3*a*e^2-5*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(e*x+d)^3/(a^2*e^4-2*a*
c*d^2*e^2+c^2*d^4)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?

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mupad [B]  time = 1.40, size = 562, normalized size = 5.06 \begin {gather*} \frac {\left (\frac {4\,c^2\,d^3}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {2\,a\,e^2}{5\,a\,e^3-5\,c\,d^2\,e}-\frac {2\,c\,d^2}{5\,a\,e^3-5\,c\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {8\,c^3\,d^4}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {\left (\frac {2\,c^2\,d^3+2\,a\,c\,d\,e^2}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {8\,c^3\,d^4}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {8\,a\,c^2\,d^2\,e}{15\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {8\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (d+e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((4*c^2*d^3)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (4*a*c*d*e^2)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*
e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((2*a*e^2)/(5*a*e^3 - 5*c*d^2*e) - (2*c*d^2
)/(5*a*e^3 - 5*c*d^2*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((4*c^3*d^4 + 4*a*c^2*d
^2*e^2)/(15*e*(a*e^2 - c*d^2)^3) - (8*c^3*d^4)/(15*e*(a*e^2 - c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^
2)^(1/2))/(d + e*x) + (((2*c^2*d^3 + 2*a*c*d*e^2)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (4*c^2*d^3)/(5*(
a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((8*c^3*d
^4)/(15*e*(a*e^2 - c*d^2)^3) - (8*a*c^2*d^2*e)/(15*(a*e^2 - c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)
^(1/2))/(d + e*x) + (8*c^2*d^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(15*e*(a*e^2 - c*d^2)^2*(d + e*x
))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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