3.17.33 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{\sqrt {d+e x}} \, dx\) [1633]

Optimal. Leaf size=127 \[ \frac {2 (b d-a e)^4 \sqrt {d+e x}}{e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{3/2}}{3 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^4 (d+e x)^{9/2}}{9 e^5} \]

[Out]

-8/3*b*(-a*e+b*d)^3*(e*x+d)^(3/2)/e^5+12/5*b^2*(-a*e+b*d)^2*(e*x+d)^(5/2)/e^5-8/7*b^3*(-a*e+b*d)*(e*x+d)^(7/2)
/e^5+2/9*b^4*(e*x+d)^(9/2)/e^5+2*(-a*e+b*d)^4*(e*x+d)^(1/2)/e^5

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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \begin {gather*} -\frac {8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac {12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac {8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac {2 \sqrt {d+e x} (b d-a e)^4}{e^5}+\frac {2 b^4 (d+e x)^{9/2}}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^4*Sqrt[d + e*x])/e^5 - (8*b*(b*d - a*e)^3*(d + e*x)^(3/2))/(3*e^5) + (12*b^2*(b*d - a*e)^2*(d +
e*x)^(5/2))/(5*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^5) + (2*b^4*(d + e*x)^(9/2))/(9*e^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^4}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 \sqrt {d+e x}}-\frac {4 b (b d-a e)^3 \sqrt {d+e x}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{5/2}}{e^4}+\frac {b^4 (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^4 \sqrt {d+e x}}{e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{3/2}}{3 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^4 (d+e x)^{9/2}}{9 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 153, normalized size = 1.20 \begin {gather*} \frac {2 \sqrt {d+e x} \left (315 a^4 e^4+420 a^3 b e^3 (-2 d+e x)+126 a^2 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+36 a b^3 e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^4 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(315*a^4*e^4 + 420*a^3*b*e^3*(-2*d + e*x) + 126*a^2*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 3
6*a*b^3*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + b^4*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*
e^3*x^3 + 35*e^4*x^4)))/(315*e^5)

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Maple [A]
time = 0.64, size = 166, normalized size = 1.31

method	result	size
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (2 a b e -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(166\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (2 a b e -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(166\)
gosper	\(\frac {2 \left (35 b^{4} x^{4} e^{4}+180 a \,b^{3} e^{4} x^{3}-40 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}-216 a \,b^{3} d \,e^{3} x^{2}+48 b^{4} d^{2} e^{2} x^{2}+420 a^{3} b \,e^{4} x -504 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x -64 b^{4} d^{3} e x +315 e^{4} a^{4}-840 a^{3} b d \,e^{3}+1008 a^{2} b^{2} d^{2} e^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (35 b^{4} x^{4} e^{4}+180 a \,b^{3} e^{4} x^{3}-40 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}-216 a \,b^{3} d \,e^{3} x^{2}+48 b^{4} d^{2} e^{2} x^{2}+420 a^{3} b \,e^{4} x -504 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x -64 b^{4} d^{3} e x +315 e^{4} a^{4}-840 a^{3} b d \,e^{3}+1008 a^{2} b^{2} d^{2} e^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\)	\(186\)
risch	\(\frac {2 \left (35 b^{4} x^{4} e^{4}+180 a \,b^{3} e^{4} x^{3}-40 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}-216 a \,b^{3} d \,e^{3} x^{2}+48 b^{4} d^{2} e^{2} x^{2}+420 a^{3} b \,e^{4} x -504 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x -64 b^{4} d^{3} e x +315 e^{4} a^{4}-840 a^{3} b d \,e^{3}+1008 a^{2} b^{2} d^{2} e^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^5*(1/9*b^4*(e*x+d)^(9/2)+2/7*(2*a*b*e-2*b^2*d)*b^2*(e*x+d)^(7/2)+1/5*(2*(a^2*e^2-2*a*b*d*e+b^2*d^2)*b^2+(2
*a*b*e-2*b^2*d)^2)*(e*x+d)^(5/2)+2/3*(a^2*e^2-2*a*b*d*e+b^2*d^2)*(2*a*b*e-2*b^2*d)*(e*x+d)^(3/2)+(a^2*e^2-2*a*
b*d*e+b^2*d^2)^2*(e*x+d)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (113) = 226\).
time = 0.28, size = 259, normalized size = 2.04 \begin {gather*} \frac {2}{315} \, {\left (84 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b^{2} e^{\left (-2\right )} + 36 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{3} e^{\left (-3\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{4} e^{\left (-4\right )} + 315 \, \sqrt {x e + d} a^{4} + 42 \, {\left (10 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b e^{\left (-1\right )} + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )}\right )} a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(84*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*e^(-2) + 36*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b^3*e^(-3) + (35*(x*e + d)^(9/2
) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^4*e^(
-4) + 315*sqrt(x*e + d)*a^4 + 42*(10*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*e^(-1) + (3*(x*e + d)^(5/2) - 1
0*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*e^(-2))*a^2)*e^(-1)

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Fricas [A]
time = 3.22, size = 164, normalized size = 1.29 \begin {gather*} \frac {2}{315} \, {\left (128 \, b^{4} d^{4} + {\left (35 \, b^{4} x^{4} + 180 \, a b^{3} x^{3} + 378 \, a^{2} b^{2} x^{2} + 420 \, a^{3} b x + 315 \, a^{4}\right )} e^{4} - 8 \, {\left (5 \, b^{4} d x^{3} + 27 \, a b^{3} d x^{2} + 63 \, a^{2} b^{2} d x + 105 \, a^{3} b d\right )} e^{3} + 48 \, {\left (b^{4} d^{2} x^{2} + 6 \, a b^{3} d^{2} x + 21 \, a^{2} b^{2} d^{2}\right )} e^{2} - 64 \, {\left (b^{4} d^{3} x + 9 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(128*b^4*d^4 + (35*b^4*x^4 + 180*a*b^3*x^3 + 378*a^2*b^2*x^2 + 420*a^3*b*x + 315*a^4)*e^4 - 8*(5*b^4*d*x
^3 + 27*a*b^3*d*x^2 + 63*a^2*b^2*d*x + 105*a^3*b*d)*e^3 + 48*(b^4*d^2*x^2 + 6*a*b^3*d^2*x + 21*a^2*b^2*d^2)*e^
2 - 64*(b^4*d^3*x + 9*a*b^3*d^3)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (117) = 234\).
time = 27.85, size = 561, normalized size = 4.42 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{4} d}{\sqrt {d + e x}} - 2 a^{4} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {8 a^{3} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {8 a^{3} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {12 a^{2} b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {12 a^{2} b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {8 a b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {8 a b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 b^{4} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 b^{4} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}}}{e} & \text {for}\: e \neq 0 \\\frac {a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a**4*d/sqrt(d + e*x) - 2*a**4*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 8*a**3*b*d*(-d/sqrt(d + e*x)
- sqrt(d + e*x))/e - 8*a**3*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 12*a**2*b**2*
d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 12*a**2*b**2*(-d**3/sqrt(d + e*x) - 3*d
**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 8*a*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*s
qrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 8*a*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d +
e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*b**4*d*(d**4/sqrt(d + e
*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 2*b
**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(
d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b*
*3*x**4 + b**4*x**5/5)/sqrt(d), True))

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Giac [A]
time = 1.08, size = 214, normalized size = 1.69 \begin {gather*} \frac {2}{315} \, {\left (420 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} b e^{\left (-1\right )} + 126 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b^{2} e^{\left (-2\right )} + 36 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{3} e^{\left (-3\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{4} e^{\left (-4\right )} + 315 \, \sqrt {x e + d} a^{4}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(420*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*e^(-1) + 126*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*a^2*b^2*e^(-2) + 36*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2
- 35*sqrt(x*e + d)*d^3)*a*b^3*e^(-3) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2
- 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^4*e^(-4) + 315*sqrt(x*e + d)*a^4)*e^(-1)

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Mupad [B]
time = 0.04, size = 112, normalized size = 0.88 \begin {gather*} \frac {2\,b^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^4*(d + e*x)^(9/2))/(9*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(7/2))/(7*e^5) + (2*(a*e - b*d)^4*(d + e*x)
^(1/2))/e^5 + (12*b^2*(a*e - b*d)^2*(d + e*x)^(5/2))/(5*e^5) + (8*b*(a*e - b*d)^3*(d + e*x)^(3/2))/(3*e^5)

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