\(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [1555]

Optimal result

Integrand size = 28, antiderivative size = 200 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {(b d-a e)^3 (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}+\frac {b (b d-a e)^2 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac {b^3 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^4 (a+b x)} \]

[Out]

-1/5*(-a*e+b*d)^3*(e*x+d)^5*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/2*b*(-a*e+b*d)^2*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^4/(
b*x+a)-3/7*b^2*(-a*e+b*d)*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/8*b^3*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^4/(b*x
+a)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^4 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{2 e^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3}{5 e^4 (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^4 (a+b x)} \]

[In]

Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-1/5*((b*d - a*e)^3*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)) + (b*(b*d - a*e)^2*(d + e*x)^6*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)) - (3*b^2*(b*d - a*e)*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(7*e^4*(a + b*x)) + (b^3*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^4*(a + b*x))

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])

Rule 660

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

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (56 a^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right )}{280 (a+b x)} \]

[In]

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(56*a^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 28*a^2*b*x*(15*d^
4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 8*a*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*
x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^
4)))/(280*(a + b*x))

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.32

method	result	size
gosper	\(\frac {x \left (35 b^{3} e^{4} x^{7}+120 x^{6} a \,b^{2} e^{4}+160 x^{6} b^{3} d \,e^{3}+140 x^{5} a^{2} b \,e^{4}+560 x^{5} a \,b^{2} d \,e^{3}+280 x^{5} b^{3} d^{2} e^{2}+56 x^{4} e^{4} a^{3}+672 x^{4} a^{2} b d \,e^{3}+1008 x^{4} a \,b^{2} d^{2} e^{2}+224 x^{4} b^{3} d^{3} e +280 x^{3} a^{3} d \,e^{3}+1260 x^{3} a^{2} b \,d^{2} e^{2}+840 x^{3} a \,b^{2} d^{3} e +70 x^{3} b^{3} d^{4}+560 a^{3} d^{2} e^{2} x^{2}+1120 a^{2} b \,d^{3} e \,x^{2}+280 a \,b^{2} d^{4} x^{2}+560 x \,a^{3} d^{3} e +420 a^{2} b \,d^{4} x +280 d^{4} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\)	\(264\)
default	\(\frac {x \left (35 b^{3} e^{4} x^{7}+120 x^{6} a \,b^{2} e^{4}+160 x^{6} b^{3} d \,e^{3}+140 x^{5} a^{2} b \,e^{4}+560 x^{5} a \,b^{2} d \,e^{3}+280 x^{5} b^{3} d^{2} e^{2}+56 x^{4} e^{4} a^{3}+672 x^{4} a^{2} b d \,e^{3}+1008 x^{4} a \,b^{2} d^{2} e^{2}+224 x^{4} b^{3} d^{3} e +280 x^{3} a^{3} d \,e^{3}+1260 x^{3} a^{2} b \,d^{2} e^{2}+840 x^{3} a \,b^{2} d^{3} e +70 x^{3} b^{3} d^{4}+560 a^{3} d^{2} e^{2} x^{2}+1120 a^{2} b \,d^{3} e \,x^{2}+280 a \,b^{2} d^{4} x^{2}+560 x \,a^{3} d^{3} e +420 a^{2} b \,d^{4} x +280 d^{4} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b x +a \right )^{3}}\)	\(264\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} e^{4} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a \,b^{2} e^{4}+4 b^{3} d \,e^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a^{2} b \,e^{4}+12 a \,b^{2} d \,e^{3}+6 b^{3} d^{2} e^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} a^{3}+12 a^{2} b d \,e^{3}+18 a \,b^{2} d^{2} e^{2}+4 b^{3} d^{3} e \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} d \,e^{3}+18 a^{2} b \,d^{2} e^{2}+12 a \,b^{2} d^{3} e +b^{3} d^{4}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{3} d^{2} e^{2}+12 a^{2} b \,d^{3} e +3 a \,b^{2} d^{4}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} d^{3} e +3 a^{2} b \,d^{4}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{4} x \,a^{3}}{b x +a}\)	\(357\)

[In]

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/280*x*(35*b^3*e^4*x^7+120*a*b^2*e^4*x^6+160*b^3*d*e^3*x^6+140*a^2*b*e^4*x^5+560*a*b^2*d*e^3*x^5+280*b^3*d^2*
e^2*x^5+56*a^3*e^4*x^4+672*a^2*b*d*e^3*x^4+1008*a*b^2*d^2*e^2*x^4+224*b^3*d^3*e*x^4+280*a^3*d*e^3*x^3+1260*a^2
*b*d^2*e^2*x^3+840*a*b^2*d^3*e*x^3+70*b^3*d^4*x^3+560*a^3*d^2*e^2*x^2+1120*a^2*b*d^3*e*x^2+280*a*b^2*d^4*x^2+5
60*a^3*d^3*e*x+420*a^2*b*d^4*x+280*a^3*d^4)*((b*x+a)^2)^(3/2)/(b*x+a)^3

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.12 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, b^{3} e^{4} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (4 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, b^{3} d^{2} e^{2} + 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, b^{3} d^{3} e + 18 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{4} + 12 \, a b^{2} d^{3} e + 18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3}\right )} x^{4} + {\left (a b^{2} d^{4} + 4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/8*b^3*e^4*x^8 + a^3*d^4*x + 1/7*(4*b^3*d*e^3 + 3*a*b^2*e^4)*x^7 + 1/2*(2*b^3*d^2*e^2 + 4*a*b^2*d*e^3 + a^2*b
*e^4)*x^6 + 1/5*(4*b^3*d^3*e + 18*a*b^2*d^2*e^2 + 12*a^2*b*d*e^3 + a^3*e^4)*x^5 + 1/4*(b^3*d^4 + 12*a*b^2*d^3*
e + 18*a^2*b*d^2*e^2 + 4*a^3*d*e^3)*x^4 + (a*b^2*d^4 + 4*a^2*b*d^3*e + 2*a^3*d^2*e^2)*x^3 + 1/2*(3*a^2*b*d^4 +
4*a^3*d^3*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5923 vs. \(2 (141) = 282\).

Time = 1.05 (sec) , antiderivative size = 5923, normalized size of antiderivative = 29.62 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**2*e**4*x**7/8 + x**6*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**
2) + x**5*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**
2*e**2)/(6*b**2) + x**4*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b*
*2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e
**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + x**3*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*
b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b
) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b*
*3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*
a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b*
*2) + x**2*(4*a**4*d*e**3 + 24*a**3*b*d**2*e**2 + 24*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d
*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 +
16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(
5*b**2) + 4*a*b**3*d**4 - 7*a*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e*
*4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b*
*3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24
*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*
b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b))/(3*b**2) + x*(6*a**4*d**2*e**2 + 16*a*
*3*b*d**3*e + 6*a**2*b**2*d**4 - 3*a**2*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a*
*2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2)
+ 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*
b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d
*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b**2) - 5*a*(4*a**4*d*e**3 + 24*
a**3*b*d**2*e**2 + 24*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/
8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*
b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4*a*b**3*d**4 - 7*a*(
a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*
(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4
+ 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a*
*2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) +
4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b))/(3*b))/(2*b**2) + (4*a**4*d**3*e + 4*a**3*b*d**4 - 2*a**2*(4*a**4*d*e
**3 + 24*a**3*b*d**2*e**2 + 24*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b
**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13
*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4*a*b**3*d**
4 - 7*a*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**
3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**
3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11
*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)
/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b))/(3*b**2) - 3*a*(6*a**4*d**2*e**2 + 16*a**3*b*d**3*e + 6*a**2
*b**2*d**4 - 3*a**2*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*
a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e
- 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d*
*2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**
4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b**2) - 5*a*(4*a**4*d*e**3 + 24*a**3*b*d**2*e**2 + 2
4*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(
7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4
*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4*a*b**3*d**4 - 7*a*(a**4*e**4 + 16*a**3*
b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 +
4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**
3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*
a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b)
+ b**4*d**4)/(4*b))/(3*b))/(2*b))/b**2) + (a/b + x)*(a**4*d**4 - a**2*(6*a**4*d**2*e**2 + 16*a**3*b*d**3*e +
6*a**2*b**2*d**4 - 3*a**2*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8
+ 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d
**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b
**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) +
6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b**2) - 5*a*(4*a**4*d*e**3 + 24*a**3*b*d**2*e*
*2 + 24*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e
**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 +
4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4*a*b**3*d**4 - 7*a*(a**4*e**4 + 16
*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**
4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2
*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8
+ 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)
/(5*b) + b**4*d**4)/(4*b))/(3*b))/(2*b**2) - a*(4*a**4*d**3*e + 4*a**3*b*d**4 - 2*a**2*(4*a**4*d*e**3 + 24*a**
3*b*d**2*e**2 + 24*a**2*b**2*d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 +
4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**
3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4*a*b**3*d**4 - 7*a*(a**
4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17
*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 2
4*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*
b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b
**4*d**3*e)/(5*b) + b**4*d**4)/(4*b))/(3*b**2) - 3*a*(6*a**4*d**2*e**2 + 16*a**3*b*d**3*e + 6*a**2*b**2*d**4 -
3*a**2*(a**4*e**4 + 16*a**3*b*d*e**3 + 36*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**
3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**
3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11
*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)
/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4)/(4*b**2) - 5*a*(4*a**4*d*e**3 + 24*a**3*b*d**2*e**2 + 24*a**2*b**2*
d**3*e - 4*a**2*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24
*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*
b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b**2) + 4*a*b**3*d**4 - 7*a*(a**4*e**4 + 16*a**3*b*d*e**3 + 3
6*a**2*b**2*d**2*e**2 - 5*a**2*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**
3)/(7*b) + 6*b**4*d**2*e**2)/(6*b**2) + 16*a*b**3*d**3*e - 9*a*(4*a**3*b*e**4 + 24*a**2*b**2*d*e**3 - 6*a**2*(
17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b**2) + 24*a*b**3*d**2*e**2 - 11*a*(41*a**2*b**2*e**4/8 + 16*a*b**3*d*e**
3 - 13*a*(17*a*b**3*e**4/8 + 4*b**4*d*e**3)/(7*b) + 6*b**4*d**2*e**2)/(6*b) + 4*b**4*d**3*e)/(5*b) + b**4*d**4
)/(4*b))/(3*b))/(2*b))/b)*log(a/b + x)/sqrt(b**2*(a/b + x)**2), Ne(b**2, 0)), (((a**2 + 2*a*b*x)**(5/2)*(a**4*
e**4 - 8*a**3*b*d*e**3 + 24*a**2*b**2*d**2*e**2 - 32*a*b**3*d**3*e + 16*b**4*d**4)/(80*b**4) + (a**2 + 2*a*b*x
)**(7/2)*(-a**3*e**4 + 6*a**2*b*d*e**3 - 12*a*b**2*d**2*e**2 + 8*b**3*d**3*e)/(28*a*b**4) + (a**2 + 2*a*b*x)**
(9/2)*(3*a**2*e**4 - 12*a*b*d*e**3 + 12*b**2*d**2*e**2)/(72*a**2*b**4) + (a**2 + 2*a*b*x)**(11/2)*(-a*e**4 + 2
*b*d*e**3)/(44*a**3*b**4) + e**4*(a**2 + 2*a*b*x)**(13/2)/(208*a**4*b**4))/(a*b), Ne(a*b, 0)), ((a**2)**(3/2)*
Piecewise((d**4*x, Eq(e, 0)), ((d + e*x)**5/(5*e), True)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (148) = 296\).

Time = 0.20 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.94 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{4} x^{3}}{8 \, b^{2}} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{4} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{3} e x}{b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{2} e^{2} x}{2 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d e^{3} x}{b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} e^{4} x}{4 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e^{3} x^{2}}{7 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{4} x^{2}}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{4}}{4 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{3} e}{b^{2}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d^{2} e^{2}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} d e^{3}}{b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} e^{4}}{4 \, b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} e^{2} x}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e^{3} x}{7 \, b^{3}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{4} x}{56 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{3} e}{5 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} e^{2}}{5 \, b^{3}} + \frac {34 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e^{3}}{35 \, b^{4}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{4}}{280 \, b^{5}} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^4*x^3/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*d^4*x - (b^2*x^2 + 2*a*b
*x + a^2)^(3/2)*a*d^3*e*x/b + 3/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^2*e^2*x/b^2 - (b^2*x^2 + 2*a*b*x + a^2
)^(3/2)*a^3*d*e^3*x/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*e^4*x/b^4 + 4/7*(b^2*x^2 + 2*a*b*x + a^2)^(5
/2)*d*e^3*x^2/b^2 - 11/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^4*x^2/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*
a*d^4/b - (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^3*e/b^2 + 3/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*d^2*e^2/b^3
- (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*d*e^3/b^4 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*e^4/b^5 + (b^2*x^2 +
2*a*b*x + a^2)^(5/2)*d^2*e^2*x/b^2 - 6/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d*e^3*x/b^3 + 13/56*(b^2*x^2 + 2*a
*b*x + a^2)^(5/2)*a^2*e^4*x/b^4 + 4/5*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^3*e/b^2 - 7/5*(b^2*x^2 + 2*a*b*x + a^2
)^(5/2)*a*d^2*e^2/b^3 + 34/35*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d*e^3/b^4 - 69/280*(b^2*x^2 + 2*a*b*x + a^2)
^(5/2)*a^3*e^4/b^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (148) = 296\).

Time = 0.29 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.16 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{8} \, b^{3} e^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, b^{3} d e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, a b^{2} e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{2} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{2} d e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{2} b e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, b^{3} d^{3} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, a b^{2} d^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, a^{2} b d e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{3} e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} d^{3} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{2} b d^{2} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + a^{3} d e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b d^{3} e x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} d^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} d^{3} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} d^{4} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (70 \, a^{4} b^{4} d^{4} - 56 \, a^{5} b^{3} d^{3} e + 28 \, a^{6} b^{2} d^{2} e^{2} - 8 \, a^{7} b d e^{3} + a^{8} e^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{280 \, b^{5}} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/8*b^3*e^4*x^8*sgn(b*x + a) + 4/7*b^3*d*e^3*x^7*sgn(b*x + a) + 3/7*a*b^2*e^4*x^7*sgn(b*x + a) + b^3*d^2*e^2*x
^6*sgn(b*x + a) + 2*a*b^2*d*e^3*x^6*sgn(b*x + a) + 1/2*a^2*b*e^4*x^6*sgn(b*x + a) + 4/5*b^3*d^3*e*x^5*sgn(b*x
+ a) + 18/5*a*b^2*d^2*e^2*x^5*sgn(b*x + a) + 12/5*a^2*b*d*e^3*x^5*sgn(b*x + a) + 1/5*a^3*e^4*x^5*sgn(b*x + a)
+ 1/4*b^3*d^4*x^4*sgn(b*x + a) + 3*a*b^2*d^3*e*x^4*sgn(b*x + a) + 9/2*a^2*b*d^2*e^2*x^4*sgn(b*x + a) + a^3*d*e
^3*x^4*sgn(b*x + a) + a*b^2*d^4*x^3*sgn(b*x + a) + 4*a^2*b*d^3*e*x^3*sgn(b*x + a) + 2*a^3*d^2*e^2*x^3*sgn(b*x
+ a) + 3/2*a^2*b*d^4*x^2*sgn(b*x + a) + 2*a^3*d^3*e*x^2*sgn(b*x + a) + a^3*d^4*x*sgn(b*x + a) + 1/280*(70*a^4*
b^4*d^4 - 56*a^5*b^3*d^3*e + 28*a^6*b^2*d^2*e^2 - 8*a^7*b*d*e^3 + a^8*e^4)*sgn(b*x + a)/b^5

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]

[In]

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)