3.1969 \(\int (a+b x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=254 \[ \frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^2}{3 e^5 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^3}{2 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^4}{7 e^5 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11}}{11 e^5 (a+b x)}-\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)}{5 e^5 (a+b x)} \]

[Out]

1/7*(-a*e+b*d)^4*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-1/2*b*(-a*e+b*d)^3*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^5/(b
*x+a)+2/3*b^2*(-a*e+b*d)^2*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-2/5*b^3*(-a*e+b*d)*(e*x+d)^10*((b*x+a)^2)^(
1/2)/e^5/(b*x+a)+1/11*b^4*(e*x+d)^11*((b*x+a)^2)^(1/2)/e^5/(b*x+a)

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Rubi [A]  time = 0.30, 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} \[ \frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11}}{11 e^5 (a+b x)}-\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)}{5 e^5 (a+b x)}+\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^2}{3 e^5 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^3}{2 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^4}{7 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) - (b*(b*d - a*e)^3*(d + e*x)^8*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)) + (2*b^2*(b*d - a*e)^2*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(3*e^5*(a + b*x)) - (2*b^3*(b*d - a*e)*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) + (b^4*
(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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)^6 \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)^6 \, 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)^6 \, 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)^6}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^7}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^8}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^9}{e^4}+\frac {b^4 (d+e x)^{10}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {b (b d-a e)^3 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}+\frac {2 b^2 (b d-a e)^2 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {2 b^3 (b d-a e) (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac {b^4 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 377, normalized size = 1.48 \[ \frac {x \sqrt {(a+b x)^2} \left (330 a^4 \left (7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right )+165 a^3 b x \left (28 d^6+112 d^5 e x+210 d^4 e^2 x^2+224 d^3 e^3 x^3+140 d^2 e^4 x^4+48 d e^5 x^5+7 e^6 x^6\right )+55 a^2 b^2 x^2 \left (84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right )+11 a b^3 x^3 \left (210 d^6+1008 d^5 e x+2100 d^4 e^2 x^2+2400 d^3 e^3 x^3+1575 d^2 e^4 x^4+560 d e^5 x^5+84 e^6 x^6\right )+b^4 x^4 \left (462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right )\right )}{2310 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(330*a^4*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^5
*x^5 + e^6*x^6) + 165*a^3*b*x*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 + 48
*d*e^5*x^5 + 7*e^6*x^6) + 55*a^2*b^2*x^2*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e
^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 11*a*b^3*x^3*(210*d^6 + 1008*d^5*e*x + 2100*d^4*e^2*x^2 + 2400*d^3*e^3*
x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6) + b^4*x^4*(462*d^6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 57
75*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6)))/(2310*(a + b*x))

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fricas [B]  time = 1.25, size = 418, normalized size = 1.65 \[ \frac {1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} + {\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3*(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*
a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 + 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*
e^2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 + (b^4*d^5*e + 10*a*b^3*d^4*e^2 +
20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 +
80*a^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3*b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4
+ (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*e)*x^2

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giac [B]  time = 0.22, size = 660, normalized size = 2.60 \[ \frac {1}{11} \, b^{4} x^{11} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, b^{4} d x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, b^{4} d^{2} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, b^{4} d^{3} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, b^{4} d^{4} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{5} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, a b^{3} x^{10} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{3} \, a b^{3} d x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a b^{3} d^{2} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {80}{7} \, a b^{3} d^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a b^{3} d^{4} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {24}{5} \, a b^{3} d^{5} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a^{2} b^{2} x^{9} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{2} b^{2} d x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {90}{7} \, a^{2} b^{2} d^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{2} b^{2} d^{3} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} d^{4} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, a^{2} b^{2} d^{5} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{3} b x^{8} e^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {24}{7} \, a^{3} b d x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b d^{2} x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 16 \, a^{3} b d^{3} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{3} b d^{4} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{3} b d^{5} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, a^{4} x^{7} e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{4} d^{2} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} d^{3} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} d^{4} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{4} d^{5} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{6} x \mathrm {sgn}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/11*b^4*x^11*e^6*sgn(b*x + a) + 3/5*b^4*d*x^10*e^5*sgn(b*x + a) + 5/3*b^4*d^2*x^9*e^4*sgn(b*x + a) + 5/2*b^4*
d^3*x^8*e^3*sgn(b*x + a) + 15/7*b^4*d^4*x^7*e^2*sgn(b*x + a) + b^4*d^5*x^6*e*sgn(b*x + a) + 1/5*b^4*d^6*x^5*sg
n(b*x + a) + 2/5*a*b^3*x^10*e^6*sgn(b*x + a) + 8/3*a*b^3*d*x^9*e^5*sgn(b*x + a) + 15/2*a*b^3*d^2*x^8*e^4*sgn(b
*x + a) + 80/7*a*b^3*d^3*x^7*e^3*sgn(b*x + a) + 10*a*b^3*d^4*x^6*e^2*sgn(b*x + a) + 24/5*a*b^3*d^5*x^5*e*sgn(b
*x + a) + a*b^3*d^6*x^4*sgn(b*x + a) + 2/3*a^2*b^2*x^9*e^6*sgn(b*x + a) + 9/2*a^2*b^2*d*x^8*e^5*sgn(b*x + a) +
90/7*a^2*b^2*d^2*x^7*e^4*sgn(b*x + a) + 20*a^2*b^2*d^3*x^6*e^3*sgn(b*x + a) + 18*a^2*b^2*d^4*x^5*e^2*sgn(b*x
+ a) + 9*a^2*b^2*d^5*x^4*e*sgn(b*x + a) + 2*a^2*b^2*d^6*x^3*sgn(b*x + a) + 1/2*a^3*b*x^8*e^6*sgn(b*x + a) + 24
/7*a^3*b*d*x^7*e^5*sgn(b*x + a) + 10*a^3*b*d^2*x^6*e^4*sgn(b*x + a) + 16*a^3*b*d^3*x^5*e^3*sgn(b*x + a) + 15*a
^3*b*d^4*x^4*e^2*sgn(b*x + a) + 8*a^3*b*d^5*x^3*e*sgn(b*x + a) + 2*a^3*b*d^6*x^2*sgn(b*x + a) + 1/7*a^4*x^7*e^
6*sgn(b*x + a) + a^4*d*x^6*e^5*sgn(b*x + a) + 3*a^4*d^2*x^5*e^4*sgn(b*x + a) + 5*a^4*d^3*x^4*e^3*sgn(b*x + a)
+ 5*a^4*d^4*x^3*e^2*sgn(b*x + a) + 3*a^4*d^5*x^2*e*sgn(b*x + a) + a^4*d^6*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 489, normalized size = 1.93 \[ \frac {\left (210 b^{4} e^{6} x^{10}+924 x^{9} a \,b^{3} e^{6}+1386 x^{9} b^{4} d \,e^{5}+1540 x^{8} a^{2} b^{2} e^{6}+6160 x^{8} a \,b^{3} d \,e^{5}+3850 x^{8} b^{4} d^{2} e^{4}+1155 x^{7} a^{3} b \,e^{6}+10395 x^{7} a^{2} b^{2} d \,e^{5}+17325 x^{7} a \,b^{3} d^{2} e^{4}+5775 x^{7} b^{4} d^{3} e^{3}+330 x^{6} a^{4} e^{6}+7920 x^{6} a^{3} b d \,e^{5}+29700 x^{6} a^{2} b^{2} d^{2} e^{4}+26400 x^{6} a \,b^{3} d^{3} e^{3}+4950 x^{6} b^{4} d^{4} e^{2}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 x^{4} a^{4} d^{2} e^{4}+36960 x^{4} a^{3} b \,d^{3} e^{3}+41580 x^{4} a^{2} b^{2} d^{4} e^{2}+11088 x^{4} a \,b^{3} d^{5} e +462 x^{4} b^{4} d^{6}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 a^{4} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{2310 \left (b x +a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/2310*x*(210*b^4*e^6*x^10+924*a*b^3*e^6*x^9+1386*b^4*d*e^5*x^9+1540*a^2*b^2*e^6*x^8+6160*a*b^3*d*e^5*x^8+3850
*b^4*d^2*e^4*x^8+1155*a^3*b*e^6*x^7+10395*a^2*b^2*d*e^5*x^7+17325*a*b^3*d^2*e^4*x^7+5775*b^4*d^3*e^3*x^7+330*a
^4*e^6*x^6+7920*a^3*b*d*e^5*x^6+29700*a^2*b^2*d^2*e^4*x^6+26400*a*b^3*d^3*e^3*x^6+4950*b^4*d^4*e^2*x^6+2310*a^
4*d*e^5*x^5+23100*a^3*b*d^2*e^4*x^5+46200*a^2*b^2*d^3*e^3*x^5+23100*a*b^3*d^4*e^2*x^5+2310*b^4*d^5*e*x^5+6930*
a^4*d^2*e^4*x^4+36960*a^3*b*d^3*e^3*x^4+41580*a^2*b^2*d^4*e^2*x^4+11088*a*b^3*d^5*e*x^4+462*b^4*d^6*x^4+11550*
a^4*d^3*e^3*x^3+34650*a^3*b*d^4*e^2*x^3+20790*a^2*b^2*d^5*e*x^3+2310*a*b^3*d^6*x^3+11550*a^4*d^4*e^2*x^2+18480
*a^3*b*d^5*e*x^2+4620*a^2*b^2*d^6*x^2+6930*a^4*d^5*e*x+4620*a^3*b*d^6*x+2310*a^4*d^6)*((b*x+a)^2)^(3/2)/(b*x+a
)^3

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maxima [B]  time = 0.64, size = 1736, normalized size = 6.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/11*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^6*x^6/b - 17/110*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^6*x^5/b^2 + 13/66*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^6*x^4/b^3 - 59/264*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^6*x^3/b^4 + 1/4
*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^6*x - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^7*e^6*x/b^6 + 21/88*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*a^4*e^6*x^2/b^5 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^6/b - 1/4*(b^2*x^2 + 2*a*b*
x + a^2)^(3/2)*a^8*e^6/b^7 - 65/264*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^6*x/b^6 + 329/1320*(b^2*x^2 + 2*a*b*
x + a^2)^(5/2)*a^6*e^6/b^7 + 1/10*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^5/b^2 - 1/6*(6*b*d*e^5
+ a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^4/b^3 + 1/3*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^
(5/2)*x^4/b^2 + 5/24*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^3/b^4 - 13/24*(5*b*d^2*e^4 + 2*
a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^3/b^3 + 5/8*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)
^(5/2)*x^3/b^2 + 1/4*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^6*x/b^6 - 3/4*(5*b*d^2*e^4 + 2*a*d*
e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*x/b^5 + 5/4*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/
2)*a^4*x/b^4 - 5/4*(3*b*d^4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 3/4*(2*b*d^5*e + 5*
a*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(b*d^6 + 6*a*d^5*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)
*a*x/b - 13/56*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x^2/b^5 + 37/56*(5*b*d^2*e^4 + 2*a*d*e^
5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^2/b^4 - 55/56*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(
5/2)*a*x^2/b^3 + 5/7*(3*b*d^4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^2/b^2 + 1/4*(6*b*d*e^5 + a*
e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^7/b^7 - 3/4*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a
^6/b^6 + 5/4*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5/b^5 - 5/4*(3*b*d^4*e^2 + 4*a*d^3*
e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^4 + 3/4*(2*b*d^5*e + 5*a*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a
^3/b^3 - 1/4*(b*d^6 + 6*a*d^5*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 + 41/168*(6*b*d*e^5 + a*e^6)*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*a^4*x/b^6 - 121/168*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^
5 + 65/56*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^4 - 15/14*(3*b*d^4*e^2 + 4*a*d^3
*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b^3 + 1/2*(2*b*d^5*e + 5*a*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*
x/b^2 - 209/840*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^7 + 125/168*(5*b*d^2*e^4 + 2*a*d*e^5
)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^6 - 69/56*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*
a^3/b^5 + 17/14*(3*b*d^4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/10*(2*b*d^5*e + 5*a*d^
4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a/b^3 + 1/5*(b*d^6 + 6*a*d^5*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 \[ \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^6\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((a + b*x)*(d + e*x)^6*(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 \[ \int \left (a + b x\right ) \left (d + e x\right )^{6} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**6*((a + b*x)**2)**(3/2), x)

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