3.1554 \(\int (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=200 \[ -\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)}{8 e^4 (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2}{7 e^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3}{6 e^4 (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^4 (a+b x)} \]

[Out]

-1/6*(-a*e+b*d)^3*(e*x+d)^6*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+3/7*b*(-a*e+b*d)^2*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^4/(
b*x+a)-3/8*b^2*(-a*e+b*d)*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^4/(b*x+a)+1/9*b^3*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^4/(b*x
+a)

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Rubi [A]  time = 0.20, antiderivative size = 200, 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 = {646, 43} \[ \frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^4 (a+b x)}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)}{8 e^4 (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2}{7 e^4 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3}{6 e^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-((b*d - a*e)^3*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^4*(a + b*x)) + (3*b*(b*d - a*e)^2*(d + e*x)^7*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)) - (3*b^2*(b*d - a*e)*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(8*e^4*(a + b*x)) + (b^3*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^4*(a + b*x))

Rule 43

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 646

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

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Mathematica [A]  time = 0.09, size = 259, normalized size = 1.30 \[ \frac {x \sqrt {(a+b x)^2} \left (84 a^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right )}{504 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(84*a^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) +
36*a^2*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*b^2*x^2*(
56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + b^3*x^3*(126*d^5 + 50
4*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/(504*(a + b*x))

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fricas [A]  time = 1.01, size = 277, normalized size = 1.38 \[ \frac {1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} + {\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/9*b^3*e^5*x^9 + a^3*d^5*x + 1/8*(5*b^3*d*e^4 + 3*a*b^2*e^5)*x^8 + 1/7*(10*b^3*d^2*e^3 + 15*a*b^2*d*e^4 + 3*a
^2*b*e^5)*x^7 + 1/6*(10*b^3*d^3*e^2 + 30*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 + a^3*e^5)*x^6 + (b^3*d^4*e + 6*a*b^2*
d^3*e^2 + 6*a^2*b*d^2*e^3 + a^3*d*e^4)*x^5 + 1/4*(b^3*d^5 + 15*a*b^2*d^4*e + 30*a^2*b*d^3*e^2 + 10*a^3*d^2*e^3
)*x^4 + 1/3*(3*a*b^2*d^5 + 15*a^2*b*d^4*e + 10*a^3*d^3*e^2)*x^3 + 1/2*(3*a^2*b*d^5 + 5*a^3*d^4*e)*x^2

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giac [B]  time = 0.18, size = 435, normalized size = 2.18 \[ \frac {1}{9} \, b^{3} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, b^{3} d x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, b^{3} d^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, b^{3} d^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{4} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, a b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{4} \, a b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, a^{2} b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{3} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{3} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{3} d^{5} x \mathrm {sgn}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/9*b^3*x^9*e^5*sgn(b*x + a) + 5/8*b^3*d*x^8*e^4*sgn(b*x + a) + 10/7*b^3*d^2*x^7*e^3*sgn(b*x + a) + 5/3*b^3*d^
3*x^6*e^2*sgn(b*x + a) + b^3*d^4*x^5*e*sgn(b*x + a) + 1/4*b^3*d^5*x^4*sgn(b*x + a) + 3/8*a*b^2*x^8*e^5*sgn(b*x
+ a) + 15/7*a*b^2*d*x^7*e^4*sgn(b*x + a) + 5*a*b^2*d^2*x^6*e^3*sgn(b*x + a) + 6*a*b^2*d^3*x^5*e^2*sgn(b*x + a
) + 15/4*a*b^2*d^4*x^4*e*sgn(b*x + a) + a*b^2*d^5*x^3*sgn(b*x + a) + 3/7*a^2*b*x^7*e^5*sgn(b*x + a) + 5/2*a^2*
b*d*x^6*e^4*sgn(b*x + a) + 6*a^2*b*d^2*x^5*e^3*sgn(b*x + a) + 15/2*a^2*b*d^3*x^4*e^2*sgn(b*x + a) + 5*a^2*b*d^
4*x^3*e*sgn(b*x + a) + 3/2*a^2*b*d^5*x^2*sgn(b*x + a) + 1/6*a^3*x^6*e^5*sgn(b*x + a) + a^3*d*x^5*e^4*sgn(b*x +
a) + 5/2*a^3*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^3*d^3*x^3*e^2*sgn(b*x + a) + 5/2*a^3*d^4*x^2*e*sgn(b*x + a) +
a^3*d^5*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 322, normalized size = 1.61 \[ \frac {\left (56 b^{3} e^{5} x^{8}+189 x^{7} a \,b^{2} e^{5}+315 x^{7} b^{3} d \,e^{4}+216 x^{6} a^{2} b \,e^{5}+1080 x^{6} a \,b^{2} d \,e^{4}+720 x^{6} b^{3} d^{2} e^{3}+84 x^{5} a^{3} e^{5}+1260 x^{5} a^{2} b d \,e^{4}+2520 x^{5} a \,b^{2} d^{2} e^{3}+840 x^{5} b^{3} d^{3} e^{2}+504 a^{3} d \,e^{4} x^{4}+3024 a^{2} b \,d^{2} e^{3} x^{4}+3024 a \,b^{2} d^{3} e^{2} x^{4}+504 b^{3} d^{4} e \,x^{4}+1260 x^{3} a^{3} d^{2} e^{3}+3780 x^{3} a^{2} b \,d^{3} e^{2}+1890 x^{3} a \,b^{2} d^{4} e +126 x^{3} b^{3} d^{5}+1680 x^{2} a^{3} d^{3} e^{2}+2520 x^{2} a^{2} b \,d^{4} e +504 x^{2} a \,b^{2} d^{5}+1260 x \,a^{3} d^{4} e +756 x \,a^{2} b \,d^{5}+504 a^{3} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{504 \left (b x +a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/504*x*(56*b^3*e^5*x^8+189*a*b^2*e^5*x^7+315*b^3*d*e^4*x^7+216*a^2*b*e^5*x^6+1080*a*b^2*d*e^4*x^6+720*b^3*d^2
*e^3*x^6+84*a^3*e^5*x^5+1260*a^2*b*d*e^4*x^5+2520*a*b^2*d^2*e^3*x^5+840*b^3*d^3*e^2*x^5+504*a^3*d*e^4*x^4+3024
*a^2*b*d^2*e^3*x^4+3024*a*b^2*d^3*e^2*x^4+504*b^3*d^4*e*x^4+1260*a^3*d^2*e^3*x^3+3780*a^2*b*d^3*e^2*x^3+1890*a
*b^2*d^4*e*x^3+126*b^3*d^5*x^3+1680*a^3*d^3*e^2*x^2+2520*a^2*b*d^4*e*x^2+504*a*b^2*d^5*x^2+1260*a^3*d^4*e*x+75
6*a^2*b*d^5*x+504*a^3*d^5)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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maxima [B]  time = 1.09, size = 814, normalized size = 4.07 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{5} x^{4}}{9 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e^{4} x^{3}}{8 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{5} x^{3}}{72 \, b^{3}} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{5} x - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{4} e x}{4 \, b} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{3} e^{2} x}{2 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d^{2} e^{3} x}{2 \, b^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} d e^{4} x}{4 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} e^{5} x}{4 \, b^{5}} + \frac {10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} e^{3} x^{2}}{7 \, b^{2}} - \frac {55 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e^{4} x^{2}}{56 \, b^{3}} + \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{5} x^{2}}{168 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{5}}{4 \, b} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{4} e}{4 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d^{3} e^{2}}{2 \, b^{3}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} d^{2} e^{3}}{2 \, b^{4}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} d e^{4}}{4 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{6} e^{5}}{4 \, b^{6}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{3} e^{2} x}{3 \, b^{2}} - \frac {15 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} e^{3} x}{7 \, b^{3}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e^{4} x}{56 \, b^{4}} - \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{5} x}{504 \, b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{4} e}{b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} e^{2}}{3 \, b^{3}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} e^{3}}{7 \, b^{4}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d e^{4}}{56 \, b^{5}} + \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{5}}{504 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/9*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^5*x^4/b^2 + 5/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d*e^4*x^3/b^2 - 13/72*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^5*x^3/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*d^5*x - 5/4*(b^2*x^2 + 2*a*b
*x + a^2)^(3/2)*a*d^4*e*x/b + 5/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^3*e^2*x/b^2 - 5/2*(b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*a^3*d^2*e^3*x/b^3 + 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*d*e^4*x/b^4 - 1/4*(b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*a^5*e^5*x/b^5 + 10/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^2*e^3*x^2/b^2 - 55/56*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*a*d*e^4*x^2/b^3 + 37/168*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^5*x^2/b^4 + 1/4*(b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*a*d^5/b - 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^4*e/b^2 + 5/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*
a^3*d^3*e^2/b^3 - 5/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*d^2*e^3/b^4 + 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^
5*d*e^4/b^5 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^6*e^5/b^6 + 5/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^3*e^2*x/
b^2 - 15/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^2*e^3*x/b^3 + 65/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d*e^4*x
/b^4 - 121/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^5*x/b^5 + (b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^4*e/b^2 - 7/3
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3*e^2/b^3 + 17/7*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^2*e^3/b^4 - 69/56*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*d*e^4/b^5 + 125/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^5/b^6

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+e\,x\right )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x\right )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((d + e*x)**5*((a + b*x)**2)**(3/2), x)

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