3.18.43 \(\int (a+b x) (d+e x)^7 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=254 \[ \frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)} \]

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Rubi [A]  time = 0.35, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)}+\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)) - (4*b*(b*d - a*e)^3*(d + e*x)^9*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (3*b^2*(b*d - a*e)^2*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(5*e^5*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (
b^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^5*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, 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 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int (a+b x) (d+e x)^7 \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 (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^7 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 (d+e x)^7 \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4 (d+e x)^7}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^8}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^9}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{10}}{e^4}+\frac {b^4 (d+e x)^{11}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {b^4 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 432, normalized size = 1.70 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (495 a^4 \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+220 a^3 b x \left (36 d^7+168 d^6 e x+378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+63 d e^6 x^6+8 e^7 x^7\right )+66 a^2 b^2 x^2 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+12 a b^3 x^3 \left (330 d^7+1848 d^6 e x+4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+924 d e^6 x^6+120 e^7 x^7\right )+b^4 x^4 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )\right )}{3960 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(495*a^4*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2*
e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 220*a^3*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 420
*d^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 66*a^2*b^2*x^2*(120*d^7 + 630*d^6*e*x + 1512*d^5*
e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7) + 12*a*b^3*x^3*(
330*d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^2 + 6600*d^4*e^3*x^3 + 5775*d^3*e^4*x^4 + 3080*d^2*e^5*x^5 + 924*d*e^6
*x^6 + 120*e^7*x^7) + b^4*x^4*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 + 15400*d^3*e^4*
x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7)))/(3960*(a + b*x))

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IntegrateAlgebraic [F]  time = 4.47, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2), x]

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fricas [B]  time = 0.40, size = 489, normalized size = 1.93 \begin {gather*} \frac {1}{12} \, b^{4} e^{7} x^{12} + a^{4} d^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} d e^{6} + 4 \, a b^{3} e^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} d^{2} e^{5} + 28 \, a b^{3} d e^{6} + 6 \, a^{2} b^{2} e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} d^{3} e^{4} + 84 \, a b^{3} d^{2} e^{5} + 42 \, a^{2} b^{2} d e^{6} + 4 \, a^{3} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} d^{4} e^{3} + 140 \, a b^{3} d^{3} e^{4} + 126 \, a^{2} b^{2} d^{2} e^{5} + 28 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{8} + {\left (3 \, b^{4} d^{5} e^{2} + 20 \, a b^{3} d^{4} e^{3} + 30 \, a^{2} b^{2} d^{3} e^{4} + 12 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} d^{6} e + 12 \, a b^{3} d^{5} e^{2} + 30 \, a^{2} b^{2} d^{4} e^{3} + 20 \, a^{3} b d^{3} e^{4} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{7} + 28 \, a b^{3} d^{6} e + 126 \, a^{2} b^{2} d^{5} e^{2} + 140 \, a^{3} b d^{4} e^{3} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{7} + 42 \, a^{2} b^{2} d^{6} e + 84 \, a^{3} b d^{5} e^{2} + 35 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{7} + 28 \, a^{3} b d^{6} e + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{7} + 7 \, a^{4} d^{6} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/12*b^4*e^7*x^12 + a^4*d^7*x + 1/11*(7*b^4*d*e^6 + 4*a*b^3*e^7)*x^11 + 1/10*(21*b^4*d^2*e^5 + 28*a*b^3*d*e^6
+ 6*a^2*b^2*e^7)*x^10 + 1/9*(35*b^4*d^3*e^4 + 84*a*b^3*d^2*e^5 + 42*a^2*b^2*d*e^6 + 4*a^3*b*e^7)*x^9 + 1/8*(35
*b^4*d^4*e^3 + 140*a*b^3*d^3*e^4 + 126*a^2*b^2*d^2*e^5 + 28*a^3*b*d*e^6 + a^4*e^7)*x^8 + (3*b^4*d^5*e^2 + 20*a
*b^3*d^4*e^3 + 30*a^2*b^2*d^3*e^4 + 12*a^3*b*d^2*e^5 + a^4*d*e^6)*x^7 + 7/6*(b^4*d^6*e + 12*a*b^3*d^5*e^2 + 30
*a^2*b^2*d^4*e^3 + 20*a^3*b*d^3*e^4 + 3*a^4*d^2*e^5)*x^6 + 1/5*(b^4*d^7 + 28*a*b^3*d^6*e + 126*a^2*b^2*d^5*e^2
+ 140*a^3*b*d^4*e^3 + 35*a^4*d^3*e^4)*x^5 + 1/4*(4*a*b^3*d^7 + 42*a^2*b^2*d^6*e + 84*a^3*b*d^5*e^2 + 35*a^4*d
^4*e^3)*x^4 + 1/3*(6*a^2*b^2*d^7 + 28*a^3*b*d^6*e + 21*a^4*d^5*e^2)*x^3 + 1/2*(4*a^3*b*d^7 + 7*a^4*d^6*e)*x^2

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giac [B]  time = 0.21, size = 761, normalized size = 3.00 \begin {gather*} \frac {1}{12} \, b^{4} x^{12} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{11} \, b^{4} d x^{11} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {21}{10} \, b^{4} d^{2} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{9} \, b^{4} d^{3} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{8} \, b^{4} d^{4} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, b^{4} d^{5} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{6} \, b^{4} d^{6} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{7} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{11} \, a b^{3} x^{11} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {14}{5} \, a b^{3} d x^{10} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {28}{3} \, a b^{3} d^{2} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{2} \, a b^{3} d^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{3} d^{4} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 14 \, a b^{3} d^{5} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {28}{5} \, a b^{3} d^{6} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{7} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a^{2} b^{2} x^{10} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {14}{3} \, a^{2} b^{2} d x^{9} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {63}{4} \, a^{2} b^{2} d^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{2} d^{3} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{2} d^{4} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {126}{5} \, a^{2} b^{2} d^{5} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {21}{2} \, a^{2} b^{2} d^{6} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{7} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, a^{3} b x^{9} e^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{2} \, a^{3} b d x^{8} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{3} b d^{2} x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {70}{3} \, a^{3} b d^{3} x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{3} b d^{4} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{3} b d^{5} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {28}{3} \, a^{3} b d^{6} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{7} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, a^{4} x^{8} e^{7} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{7} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{2} \, a^{4} d^{2} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} d^{3} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {35}{4} \, a^{4} d^{4} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} d^{5} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {7}{2} \, a^{4} d^{6} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{7} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/12*b^4*x^12*e^7*sgn(b*x + a) + 7/11*b^4*d*x^11*e^6*sgn(b*x + a) + 21/10*b^4*d^2*x^10*e^5*sgn(b*x + a) + 35/9
*b^4*d^3*x^9*e^4*sgn(b*x + a) + 35/8*b^4*d^4*x^8*e^3*sgn(b*x + a) + 3*b^4*d^5*x^7*e^2*sgn(b*x + a) + 7/6*b^4*d
^6*x^6*e*sgn(b*x + a) + 1/5*b^4*d^7*x^5*sgn(b*x + a) + 4/11*a*b^3*x^11*e^7*sgn(b*x + a) + 14/5*a*b^3*d*x^10*e^
6*sgn(b*x + a) + 28/3*a*b^3*d^2*x^9*e^5*sgn(b*x + a) + 35/2*a*b^3*d^3*x^8*e^4*sgn(b*x + a) + 20*a*b^3*d^4*x^7*
e^3*sgn(b*x + a) + 14*a*b^3*d^5*x^6*e^2*sgn(b*x + a) + 28/5*a*b^3*d^6*x^5*e*sgn(b*x + a) + a*b^3*d^7*x^4*sgn(b
*x + a) + 3/5*a^2*b^2*x^10*e^7*sgn(b*x + a) + 14/3*a^2*b^2*d*x^9*e^6*sgn(b*x + a) + 63/4*a^2*b^2*d^2*x^8*e^5*s
gn(b*x + a) + 30*a^2*b^2*d^3*x^7*e^4*sgn(b*x + a) + 35*a^2*b^2*d^4*x^6*e^3*sgn(b*x + a) + 126/5*a^2*b^2*d^5*x^
5*e^2*sgn(b*x + a) + 21/2*a^2*b^2*d^6*x^4*e*sgn(b*x + a) + 2*a^2*b^2*d^7*x^3*sgn(b*x + a) + 4/9*a^3*b*x^9*e^7*
sgn(b*x + a) + 7/2*a^3*b*d*x^8*e^6*sgn(b*x + a) + 12*a^3*b*d^2*x^7*e^5*sgn(b*x + a) + 70/3*a^3*b*d^3*x^6*e^4*s
gn(b*x + a) + 28*a^3*b*d^4*x^5*e^3*sgn(b*x + a) + 21*a^3*b*d^5*x^4*e^2*sgn(b*x + a) + 28/3*a^3*b*d^6*x^3*e*sgn
(b*x + a) + 2*a^3*b*d^7*x^2*sgn(b*x + a) + 1/8*a^4*x^8*e^7*sgn(b*x + a) + a^4*d*x^7*e^6*sgn(b*x + a) + 7/2*a^4
*d^2*x^6*e^5*sgn(b*x + a) + 7*a^4*d^3*x^5*e^4*sgn(b*x + a) + 35/4*a^4*d^4*x^4*e^3*sgn(b*x + a) + 7*a^4*d^5*x^3
*e^2*sgn(b*x + a) + 7/2*a^4*d^6*x^2*e*sgn(b*x + a) + a^4*d^7*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 564, normalized size = 2.22 \begin {gather*} \frac {\left (330 b^{4} e^{7} x^{11}+1440 x^{10} a \,b^{3} e^{7}+2520 x^{10} b^{4} d \,e^{6}+2376 x^{9} a^{2} b^{2} e^{7}+11088 x^{9} a \,b^{3} d \,e^{6}+8316 x^{9} b^{4} d^{2} e^{5}+1760 x^{8} a^{3} b \,e^{7}+18480 x^{8} a^{2} b^{2} d \,e^{6}+36960 x^{8} a \,b^{3} d^{2} e^{5}+15400 x^{8} b^{4} d^{3} e^{4}+495 x^{7} a^{4} e^{7}+13860 x^{7} a^{3} b d \,e^{6}+62370 x^{7} a^{2} b^{2} d^{2} e^{5}+69300 x^{7} a \,b^{3} d^{3} e^{4}+17325 x^{7} b^{4} d^{4} e^{3}+3960 a^{4} d \,e^{6} x^{6}+47520 a^{3} b \,d^{2} e^{5} x^{6}+118800 a^{2} b^{2} d^{3} e^{4} x^{6}+79200 a \,b^{3} d^{4} e^{3} x^{6}+11880 b^{4} d^{5} e^{2} x^{6}+13860 x^{5} a^{4} d^{2} e^{5}+92400 x^{5} a^{3} b \,d^{3} e^{4}+138600 x^{5} a^{2} b^{2} d^{4} e^{3}+55440 x^{5} a \,b^{3} d^{5} e^{2}+4620 x^{5} b^{4} d^{6} e +27720 x^{4} a^{4} d^{3} e^{4}+110880 x^{4} a^{3} b \,d^{4} e^{3}+99792 x^{4} a^{2} b^{2} d^{5} e^{2}+22176 x^{4} a \,b^{3} d^{6} e +792 x^{4} b^{4} d^{7}+34650 x^{3} a^{4} d^{4} e^{3}+83160 x^{3} a^{3} b \,d^{5} e^{2}+41580 x^{3} a^{2} b^{2} d^{6} e +3960 x^{3} a \,b^{3} d^{7}+27720 x^{2} a^{4} d^{5} e^{2}+36960 x^{2} a^{3} b \,d^{6} e +7920 x^{2} a^{2} b^{2} d^{7}+13860 x \,a^{4} d^{6} e +7920 x \,a^{3} b \,d^{7}+3960 a^{4} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{3960 \left (b x +a \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/3960*x*(330*b^4*e^7*x^11+1440*a*b^3*e^7*x^10+2520*b^4*d*e^6*x^10+2376*a^2*b^2*e^7*x^9+11088*a*b^3*d*e^6*x^9+
8316*b^4*d^2*e^5*x^9+1760*a^3*b*e^7*x^8+18480*a^2*b^2*d*e^6*x^8+36960*a*b^3*d^2*e^5*x^8+15400*b^4*d^3*e^4*x^8+
495*a^4*e^7*x^7+13860*a^3*b*d*e^6*x^7+62370*a^2*b^2*d^2*e^5*x^7+69300*a*b^3*d^3*e^4*x^7+17325*b^4*d^4*e^3*x^7+
3960*a^4*d*e^6*x^6+47520*a^3*b*d^2*e^5*x^6+118800*a^2*b^2*d^3*e^4*x^6+79200*a*b^3*d^4*e^3*x^6+11880*b^4*d^5*e^
2*x^6+13860*a^4*d^2*e^5*x^5+92400*a^3*b*d^3*e^4*x^5+138600*a^2*b^2*d^4*e^3*x^5+55440*a*b^3*d^5*e^2*x^5+4620*b^
4*d^6*e*x^5+27720*a^4*d^3*e^4*x^4+110880*a^3*b*d^4*e^3*x^4+99792*a^2*b^2*d^5*e^2*x^4+22176*a*b^3*d^6*e*x^4+792
*b^4*d^7*x^4+34650*a^4*d^4*e^3*x^3+83160*a^3*b*d^5*e^2*x^3+41580*a^2*b^2*d^6*e*x^3+3960*a*b^3*d^7*x^3+27720*a^
4*d^5*e^2*x^2+36960*a^3*b*d^6*e*x^2+7920*a^2*b^2*d^7*x^2+13860*a^4*d^6*e*x+7920*a^3*b*d^7*x+3960*a^4*d^7)*((b*
x+a)^2)^(3/2)/(b*x+a)^3

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maxima [B]  time = 0.75, size = 2152, normalized size = 8.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/12*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^7*x^7/b - 19/132*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^7*x^6/b^2 + 41/220
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^7*x^5/b^3 - 85/396*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^7*x^4/b^4 + 23
/99*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^7*x^3/b^5 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^7*x + 1/4*(b^2*x
^2 + 2*a*b*x + a^2)^(3/2)*a^8*e^7*x/b^7 - 8/33*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^7*x^2/b^6 + 1/4*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*a^2*d^7/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^9*e^7/b^8 + 49/198*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a^6*e^7*x/b^7 - 247/990*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7*e^7/b^8 + 1/11*(7*b*d*e^6 + a*e^7)*(
b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^6/b^2 - 17/110*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^5/b^3
+ 7/10*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^5/b^2 + 13/66*(7*b*d*e^6 + a*e^7)*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*a^2*x^4/b^4 - 7/6*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^4/b^3 + 7/9
*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^4/b^2 - 59/264*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*a^3*x^3/b^5 + 35/24*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^3/b^4 -
91/72*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^3/b^3 + 35/8*(b*d^4*e^3 + a*d^3*e^4)*(b^
2*x^2 + 2*a*b*x + a^2)^(5/2)*x^3/b^2 - 1/4*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^7*x/b^7 + 7/4
*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^6*x/b^6 - 7/4*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*a^5*x/b^5 + 35/4*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*x/b^4 - 7/
4*(3*b*d^5*e^2 + 5*a*d^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 7/4*(b*d^6*e + 3*a*d^5*e^2)*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(b*d^7 + 7*a*d^6*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b + 21/88*(7*b
*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x^2/b^6 - 13/8*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a^3*x^2/b^5 + 37/24*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^2/b^4 - 55/
8*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^2/b^3 + (3*b*d^5*e^2 + 5*a*d^4*e^3)*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*x^2/b^2 - 1/4*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^8/b^8 + 7/4*(3*b*d^2*e
^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^7/b^7 - 7/4*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a
^2)^(3/2)*a^6/b^6 + 35/4*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5/b^5 - 7/4*(3*b*d^5*e^2 +
5*a*d^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^4 + 7/4*(b*d^6*e + 3*a*d^5*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(
3/2)*a^3/b^3 - 1/4*(b*d^7 + 7*a*d^6*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 - 65/264*(7*b*d*e^6 + a*e^7)*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*x/b^7 + 41/24*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/
b^6 - 121/72*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^5 + 65/8*(b*d^4*e^3 + a*d^3*e
^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^4 - 3/2*(3*b*d^5*e^2 + 5*a*d^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2
)*a*x/b^3 + 7/6*(b*d^6*e + 3*a*d^5*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x/b^2 + 329/1320*(7*b*d*e^6 + a*e^7)*(
b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6/b^8 - 209/120*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b
^7 + 125/72*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^6 - 69/8*(b*d^4*e^3 + a*d^3*e^4)
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^5 + 17/10*(3*b*d^5*e^2 + 5*a*d^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a
^2/b^4 - 49/30*(b*d^6*e + 3*a*d^5*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a/b^3 + 1/5*(b*d^7 + 7*a*d^6*e)*(b^2*x^
2 + 2*a*b*x + a^2)^(5/2)/b^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^7\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^7*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((a + b*x)*(d + e*x)^7*(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 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{7} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**7*((a + b*x)**2)**(3/2), x)

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