3.18.65 \(\int (a+b x) (d+e x)^7 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=362 \[ \frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^4}{2 e^7 (a+b x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^5}{3 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^6}{8 e^7 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{14}}{14 e^7 (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)}{13 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^2}{4 e^7 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^3}{11 e^7 (a+b x)} \]

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Rubi [A]  time = 0.52, antiderivative size = 362, 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^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{14}}{14 e^7 (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)}{13 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^2}{4 e^7 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^3}{11 e^7 (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)^4}{2 e^7 (a+b x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^5}{3 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^6}{8 e^7 (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)^(5/2),x]

[Out]

((b*d - a*e)^6*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)) - (2*b*(b*d - a*e)^5*(d + e*x)^9*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (3*b^2*(b*d - a*e)^4*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x))
+ (5*b^4*(b*d - a*e)^2*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) - (6*b^5*(b*d - a*e)*(d +
e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (b^6*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(14*e^7*(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 )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^7 \, dx}{b^4 \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)^6 (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)^6 (d+e x)^7}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^8}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^9}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{10}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{11}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{12}}{e^6}+\frac {b^6 (d+e x)^{13}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^6 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x)}-\frac {2 b (b d-a e)^5 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {3 b^2 (b d-a e)^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {b^6 (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 602, normalized size = 1.66 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (3003 a^6 \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 )+2002 a^5 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 )+1001 a^4 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 )+364 a^3 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 )+91 a^2 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 )+14 a b^5 x^5 \left (1716 d^7+10296 d^6 e x+27027 d^5 e^2 x^2+40040 d^4 e^3 x^3+36036 d^3 e^4 x^4+19656 d^2 e^5 x^5+6006 d e^6 x^6+792 e^7 x^7\right )+b^6 x^6 \left (3432 d^7+21021 d^6 e x+56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+12936 d e^6 x^6+1716 e^7 x^7\right )\right )}{24024 (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)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(3003*a^6*(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) + 2002*a^5*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 4
20*d^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 1001*a^4*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) + 364*a^3*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 + 92
4*d*e^6*x^6 + 120*e^7*x^7) + 91*a^2*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) + 14*a*b^5*x^5*(1716*d^7 + 10296*d^6*e*x
+ 27027*d^5*e^2*x^2 + 40040*d^4*e^3*x^3 + 36036*d^3*e^4*x^4 + 19656*d^2*e^5*x^5 + 6006*d*e^6*x^6 + 792*e^7*x^7
) + b^6*x^6*(3432*d^7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76440*d^3*e^4*x^4 + 42042*d^2*
e^5*x^5 + 12936*d*e^6*x^6 + 1716*e^7*x^7)))/(24024*(a + b*x))

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IntegrateAlgebraic [F]  time = 6.81, 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 )^{5/2} \, 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)^(5/2),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]

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fricas [B]  time = 0.44, size = 706, normalized size = 1.95 \begin {gather*} \frac {1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac {1}{13} \, {\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac {1}{4} \, {\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac {1}{11} \, {\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{7} + 7 \, a^{6} 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)^(5/2),x, algorithm="fricas")

[Out]

1/14*b^6*e^7*x^14 + a^6*d^7*x + 1/13*(7*b^6*d*e^6 + 6*a*b^5*e^7)*x^13 + 1/4*(7*b^6*d^2*e^5 + 14*a*b^5*d*e^6 +
5*a^2*b^4*e^7)*x^12 + 1/11*(35*b^6*d^3*e^4 + 126*a*b^5*d^2*e^5 + 105*a^2*b^4*d*e^6 + 20*a^3*b^3*e^7)*x^11 + 1/
2*(7*b^6*d^4*e^3 + 42*a*b^5*d^3*e^4 + 63*a^2*b^4*d^2*e^5 + 28*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + 1/3*(7*b^6
*d^5*e^2 + 70*a*b^5*d^4*e^3 + 175*a^2*b^4*d^3*e^4 + 140*a^3*b^3*d^2*e^5 + 35*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*x^9
+ 1/8*(7*b^6*d^6*e + 126*a*b^5*d^5*e^2 + 525*a^2*b^4*d^4*e^3 + 700*a^3*b^3*d^3*e^4 + 315*a^4*b^2*d^2*e^5 + 42*
a^5*b*d*e^6 + a^6*e^7)*x^8 + 1/7*(b^6*d^7 + 42*a*b^5*d^6*e + 315*a^2*b^4*d^5*e^2 + 700*a^3*b^3*d^4*e^3 + 525*a
^4*b^2*d^3*e^4 + 126*a^5*b*d^2*e^5 + 7*a^6*d*e^6)*x^7 + 1/2*(2*a*b^5*d^7 + 35*a^2*b^4*d^6*e + 140*a^3*b^3*d^5*
e^2 + 175*a^4*b^2*d^4*e^3 + 70*a^5*b*d^3*e^4 + 7*a^6*d^2*e^5)*x^6 + (3*a^2*b^4*d^7 + 28*a^3*b^3*d^6*e + 63*a^4
*b^2*d^5*e^2 + 42*a^5*b*d^4*e^3 + 7*a^6*d^3*e^4)*x^5 + 1/4*(20*a^3*b^3*d^7 + 105*a^4*b^2*d^6*e + 126*a^5*b*d^5
*e^2 + 35*a^6*d^4*e^3)*x^4 + (5*a^4*b^2*d^7 + 14*a^5*b*d^6*e + 7*a^6*d^5*e^2)*x^3 + 1/2*(6*a^5*b*d^7 + 7*a^6*d
^6*e)*x^2

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giac [B]  time = 0.28, size = 1099, normalized size = 3.04

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)^(5/2),x, algorithm="giac")

[Out]

1/14*b^6*x^14*e^7*sgn(b*x + a) + 7/13*b^6*d*x^13*e^6*sgn(b*x + a) + 7/4*b^6*d^2*x^12*e^5*sgn(b*x + a) + 35/11*
b^6*d^3*x^11*e^4*sgn(b*x + a) + 7/2*b^6*d^4*x^10*e^3*sgn(b*x + a) + 7/3*b^6*d^5*x^9*e^2*sgn(b*x + a) + 7/8*b^6
*d^6*x^8*e*sgn(b*x + a) + 1/7*b^6*d^7*x^7*sgn(b*x + a) + 6/13*a*b^5*x^13*e^7*sgn(b*x + a) + 7/2*a*b^5*d*x^12*e
^6*sgn(b*x + a) + 126/11*a*b^5*d^2*x^11*e^5*sgn(b*x + a) + 21*a*b^5*d^3*x^10*e^4*sgn(b*x + a) + 70/3*a*b^5*d^4
*x^9*e^3*sgn(b*x + a) + 63/4*a*b^5*d^5*x^8*e^2*sgn(b*x + a) + 6*a*b^5*d^6*x^7*e*sgn(b*x + a) + a*b^5*d^7*x^6*s
gn(b*x + a) + 5/4*a^2*b^4*x^12*e^7*sgn(b*x + a) + 105/11*a^2*b^4*d*x^11*e^6*sgn(b*x + a) + 63/2*a^2*b^4*d^2*x^
10*e^5*sgn(b*x + a) + 175/3*a^2*b^4*d^3*x^9*e^4*sgn(b*x + a) + 525/8*a^2*b^4*d^4*x^8*e^3*sgn(b*x + a) + 45*a^2
*b^4*d^5*x^7*e^2*sgn(b*x + a) + 35/2*a^2*b^4*d^6*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^7*x^5*sgn(b*x + a) + 20/11*a
^3*b^3*x^11*e^7*sgn(b*x + a) + 14*a^3*b^3*d*x^10*e^6*sgn(b*x + a) + 140/3*a^3*b^3*d^2*x^9*e^5*sgn(b*x + a) + 1
75/2*a^3*b^3*d^3*x^8*e^4*sgn(b*x + a) + 100*a^3*b^3*d^4*x^7*e^3*sgn(b*x + a) + 70*a^3*b^3*d^5*x^6*e^2*sgn(b*x
+ a) + 28*a^3*b^3*d^6*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^7*x^4*sgn(b*x + a) + 3/2*a^4*b^2*x^10*e^7*sgn(b*x + a)
+ 35/3*a^4*b^2*d*x^9*e^6*sgn(b*x + a) + 315/8*a^4*b^2*d^2*x^8*e^5*sgn(b*x + a) + 75*a^4*b^2*d^3*x^7*e^4*sgn(b*
x + a) + 175/2*a^4*b^2*d^4*x^6*e^3*sgn(b*x + a) + 63*a^4*b^2*d^5*x^5*e^2*sgn(b*x + a) + 105/4*a^4*b^2*d^6*x^4*
e*sgn(b*x + a) + 5*a^4*b^2*d^7*x^3*sgn(b*x + a) + 2/3*a^5*b*x^9*e^7*sgn(b*x + a) + 21/4*a^5*b*d*x^8*e^6*sgn(b*
x + a) + 18*a^5*b*d^2*x^7*e^5*sgn(b*x + a) + 35*a^5*b*d^3*x^6*e^4*sgn(b*x + a) + 42*a^5*b*d^4*x^5*e^3*sgn(b*x
+ a) + 63/2*a^5*b*d^5*x^4*e^2*sgn(b*x + a) + 14*a^5*b*d^6*x^3*e*sgn(b*x + a) + 3*a^5*b*d^7*x^2*sgn(b*x + a) +
1/8*a^6*x^8*e^7*sgn(b*x + a) + a^6*d*x^7*e^6*sgn(b*x + a) + 7/2*a^6*d^2*x^6*e^5*sgn(b*x + a) + 7*a^6*d^3*x^5*e
^4*sgn(b*x + a) + 35/4*a^6*d^4*x^4*e^3*sgn(b*x + a) + 7*a^6*d^5*x^3*e^2*sgn(b*x + a) + 7/2*a^6*d^6*x^2*e*sgn(b
*x + a) + a^6*d^7*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 816, normalized size = 2.25 \begin {gather*} \frac {\left (1716 b^{6} e^{7} x^{13}+11088 x^{12} a \,b^{5} e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} a^{2} b^{4} e^{7}+84084 x^{11} a \,b^{5} d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} a^{2} b^{4} d \,e^{6}+275184 x^{10} a \,b^{5} d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} a^{2} b^{4} d^{2} e^{5}+504504 x^{9} a \,b^{5} d^{3} e^{4}+84084 x^{9} b^{6} d^{4} e^{3}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} a^{2} b^{4} d^{3} e^{4}+560560 x^{8} a \,b^{5} d^{4} e^{3}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} a^{2} b^{4} d^{4} e^{3}+378378 x^{7} a \,b^{5} d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} d^{4} e^{3}+1081080 x^{6} a^{2} b^{4} d^{5} e^{2}+144144 x^{6} a \,b^{5} d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} d^{4} e^{3}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} a^{2} b^{4} d^{6} e +24024 x^{5} a \,b^{5} d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} d^{4} e^{3}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{24024 \left (b x +a \right )^{5}} \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)^(5/2),x)

[Out]

1/24024*x*(1716*b^6*e^7*x^13+11088*a*b^5*e^7*x^12+12936*b^6*d*e^6*x^12+30030*a^2*b^4*e^7*x^11+84084*a*b^5*d*e^
6*x^11+42042*b^6*d^2*e^5*x^11+43680*a^3*b^3*e^7*x^10+229320*a^2*b^4*d*e^6*x^10+275184*a*b^5*d^2*e^5*x^10+76440
*b^6*d^3*e^4*x^10+36036*a^4*b^2*e^7*x^9+336336*a^3*b^3*d*e^6*x^9+756756*a^2*b^4*d^2*e^5*x^9+504504*a*b^5*d^3*e
^4*x^9+84084*b^6*d^4*e^3*x^9+16016*a^5*b*e^7*x^8+280280*a^4*b^2*d*e^6*x^8+1121120*a^3*b^3*d^2*e^5*x^8+1401400*
a^2*b^4*d^3*e^4*x^8+560560*a*b^5*d^4*e^3*x^8+56056*b^6*d^5*e^2*x^8+3003*a^6*e^7*x^7+126126*a^5*b*d*e^6*x^7+945
945*a^4*b^2*d^2*e^5*x^7+2102100*a^3*b^3*d^3*e^4*x^7+1576575*a^2*b^4*d^4*e^3*x^7+378378*a*b^5*d^5*e^2*x^7+21021
*b^6*d^6*e*x^7+24024*a^6*d*e^6*x^6+432432*a^5*b*d^2*e^5*x^6+1801800*a^4*b^2*d^3*e^4*x^6+2402400*a^3*b^3*d^4*e^
3*x^6+1081080*a^2*b^4*d^5*e^2*x^6+144144*a*b^5*d^6*e*x^6+3432*b^6*d^7*x^6+84084*a^6*d^2*e^5*x^5+840840*a^5*b*d
^3*e^4*x^5+2102100*a^4*b^2*d^4*e^3*x^5+1681680*a^3*b^3*d^5*e^2*x^5+420420*a^2*b^4*d^6*e*x^5+24024*a*b^5*d^7*x^
5+168168*a^6*d^3*e^4*x^4+1009008*a^5*b*d^4*e^3*x^4+1513512*a^4*b^2*d^5*e^2*x^4+672672*a^3*b^3*d^6*e*x^4+72072*
a^2*b^4*d^7*x^4+210210*a^6*d^4*e^3*x^3+756756*a^5*b*d^5*e^2*x^3+630630*a^4*b^2*d^6*e*x^3+120120*a^3*b^3*d^7*x^
3+168168*a^6*d^5*e^2*x^2+336336*a^5*b*d^6*e*x^2+120120*a^4*b^2*d^7*x^2+84084*a^6*d^6*e*x+72072*a^5*b*d^7*x+240
24*a^6*d^7)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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maxima [B]  time = 0.72, size = 2153, normalized size = 5.95

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)^(5/2),x, algorithm="maxima")

[Out]

1/14*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^7*x^7/b - 3/26*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^7*x^6/b^2 + 11/78*(b
^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^7*x^5/b^3 - 133/858*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^7*x^4/b^4 + 139/
858*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^7*x^3/b^5 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^7*x + 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(5/2)*a^8*e^7*x/b^7 - 425/2574*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*e^7*x^2/b^6 + 1/6*(b^2*
x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^7/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9*e^7/b^8 + 214/1287*(b^2*x^2 + 2
*a*b*x + a^2)^(7/2)*a^6*e^7*x/b^7 - 1501/9009*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7*e^7/b^8 + 1/13*(7*b*d*e^6 +
a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^6/b^2 - 19/156*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*
x^5/b^3 + 7/12*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^5/b^2 + 251/1716*(7*b*d*e^6 + a*e^7)*
(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^4/b^4 - 119/132*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*
a*x^4/b^3 + 7/11*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^4/b^2 - 68/429*(7*b*d*e^6 + a*e
^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^3/b^5 + 35/33*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2
)*a^2*x^3/b^4 - 21/22*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 7/2*(b*d^4*e^3 +
a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 - 1/6*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*
a^7*x/b^7 + 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^6*x/b^6 - 7/6*(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^5*x/b^5 + 35/6*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)
*a^4*x/b^4 - 7/6*(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^3*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)*a^2*x/b^2 - 1/6*(b*d^7 + 7*a*d^6*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x
/b + 211/1287*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x^2/b^6 - 112/99*(3*b*d^2*e^5 + a*d*e^6)
*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^2/b^5 + 217/198*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(
7/2)*a^2*x^2/b^4 - 91/18*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^2/b^3 + 7/9*(3*b*d^5*e^2
+ 5*a*d^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^2/b^2 - 1/6*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/
2)*a^8/b^8 + 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^7/b^7 - 7/6*(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^6/b^6 + 35/6*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^
5/b^5 - 7/6*(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^4/b^4 + 7/6*(b*d^6*e + 3*a*d^5*e^2)*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(b*d^7 + 7*a*d^6*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 - 17
09/10296*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*x/b^7 + 917/792*(3*b*d^2*e^5 + a*d*e^6)*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x/b^6 - 455/396*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3
*x/b^5 + 203/36*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x/b^4 - 77/72*(3*b*d^5*e^2 + 5*a*d
^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 7/8*(b*d^6*e + 3*a*d^5*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*
x/b^2 + 1715/10296*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6/b^8 - 923/792*(3*b*d^2*e^5 + a*d*e^
6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5/b^7 + 461/396*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/
2)*a^4/b^6 - 209/36*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 83/72*(3*b*d^5*e^2 + 5*a
*d^4*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 9/8*(b*d^6*e + 3*a*d^5*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2
)*a/b^3 + 1/7*(b*d^7 + 7*a*d^6*e)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/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 )}^{5/2} \,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)^(5/2),x)

[Out]

int((a + b*x)*(d + e*x)^7*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), 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 {5}{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)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**7*((a + b*x)**2)**(5/2), x)

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