3.18.63 \(\int (a+b x) (d+e x)^9 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)
Optimal. Leaf size=362 \[ \frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^4}{4 e^7 (a+b x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^5}{11 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^6}{10 e^7 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{16}}{16 e^7 (a+b x)}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15} (b d-a e)}{5 e^7 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)^2}{14 e^7 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^3}{13 e^7 (a+b x)} \]
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Rubi [A] time = 0.69, 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)^{16}}{16 e^7 (a+b x)}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15} (b d-a e)}{5 e^7 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)^2}{14 e^7 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^3}{13 e^7 (a+b x)}+\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^4}{4 e^7 (a+b x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^5}{11 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^6}{10 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)*(d + e*x)^9*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
((b*d - a*e)^6*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7*(a + b*x)) - (6*b*(b*d - a*e)^5*(d + e*x)^1
1*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (5*b^2*(b*d - a*e)^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(4*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*
x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)) - (2*b^5*(b*d - a*e
)*(d + e*x)^15*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (b^6*(d + e*x)^16*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(16*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)^9 \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)^9 \, 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)^9 \, 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)^9}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{10}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{12}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{14}}{e^6}+\frac {b^6 (d+e x)^{15}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^6 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x)}-\frac {6 b (b d-a e)^5 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {5 b^2 (b d-a e)^4 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)}-\frac {2 b^5 (b d-a e) (d+e x)^{15} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {b^6 (d+e x)^{16} \sqrt {a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x)}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 756, normalized size = 2.09 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (8008 a^6 \left (10 d^9+45 d^8 e x+120 d^7 e^2 x^2+210 d^6 e^3 x^3+252 d^5 e^4 x^4+210 d^4 e^5 x^5+120 d^3 e^6 x^6+45 d^2 e^7 x^7+10 d e^8 x^8+e^9 x^9\right )+4368 a^5 b x \left (55 d^9+330 d^8 e x+990 d^7 e^2 x^2+1848 d^6 e^3 x^3+2310 d^5 e^4 x^4+1980 d^4 e^5 x^5+1155 d^3 e^6 x^6+440 d^2 e^7 x^7+99 d e^8 x^8+10 e^9 x^9\right )+1820 a^4 b^2 x^2 \left (220 d^9+1485 d^8 e x+4752 d^7 e^2 x^2+9240 d^6 e^3 x^3+11880 d^5 e^4 x^4+10395 d^4 e^5 x^5+6160 d^3 e^6 x^6+2376 d^2 e^7 x^7+540 d e^8 x^8+55 e^9 x^9\right )+560 a^3 b^3 x^3 \left (715 d^9+5148 d^8 e x+17160 d^7 e^2 x^2+34320 d^6 e^3 x^3+45045 d^5 e^4 x^4+40040 d^4 e^5 x^5+24024 d^3 e^6 x^6+9360 d^2 e^7 x^7+2145 d e^8 x^8+220 e^9 x^9\right )+120 a^2 b^4 x^4 \left (2002 d^9+15015 d^8 e x+51480 d^7 e^2 x^2+105105 d^6 e^3 x^3+140140 d^5 e^4 x^4+126126 d^4 e^5 x^5+76440 d^3 e^6 x^6+30030 d^2 e^7 x^7+6930 d e^8 x^8+715 e^9 x^9\right )+16 a b^5 x^5 \left (5005 d^9+38610 d^8 e x+135135 d^7 e^2 x^2+280280 d^6 e^3 x^3+378378 d^5 e^4 x^4+343980 d^4 e^5 x^5+210210 d^3 e^6 x^6+83160 d^2 e^7 x^7+19305 d e^8 x^8+2002 e^9 x^9\right )+b^6 x^6 \left (11440 d^9+90090 d^8 e x+320320 d^7 e^2 x^2+672672 d^6 e^3 x^3+917280 d^5 e^4 x^4+840840 d^4 e^5 x^5+517440 d^3 e^6 x^6+205920 d^2 e^7 x^7+48048 d e^8 x^8+5005 e^9 x^9\right )\right )}{80080 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)*(d + e*x)^9*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
(x*Sqrt[(a + b*x)^2]*(8008*a^6*(10*d^9 + 45*d^8*e*x + 120*d^7*e^2*x^2 + 210*d^6*e^3*x^3 + 252*d^5*e^4*x^4 + 21
0*d^4*e^5*x^5 + 120*d^3*e^6*x^6 + 45*d^2*e^7*x^7 + 10*d*e^8*x^8 + e^9*x^9) + 4368*a^5*b*x*(55*d^9 + 330*d^8*e*
x + 990*d^7*e^2*x^2 + 1848*d^6*e^3*x^3 + 2310*d^5*e^4*x^4 + 1980*d^4*e^5*x^5 + 1155*d^3*e^6*x^6 + 440*d^2*e^7*
x^7 + 99*d*e^8*x^8 + 10*e^9*x^9) + 1820*a^4*b^2*x^2*(220*d^9 + 1485*d^8*e*x + 4752*d^7*e^2*x^2 + 9240*d^6*e^3*
x^3 + 11880*d^5*e^4*x^4 + 10395*d^4*e^5*x^5 + 6160*d^3*e^6*x^6 + 2376*d^2*e^7*x^7 + 540*d*e^8*x^8 + 55*e^9*x^9
) + 560*a^3*b^3*x^3*(715*d^9 + 5148*d^8*e*x + 17160*d^7*e^2*x^2 + 34320*d^6*e^3*x^3 + 45045*d^5*e^4*x^4 + 4004
0*d^4*e^5*x^5 + 24024*d^3*e^6*x^6 + 9360*d^2*e^7*x^7 + 2145*d*e^8*x^8 + 220*e^9*x^9) + 120*a^2*b^4*x^4*(2002*d
^9 + 15015*d^8*e*x + 51480*d^7*e^2*x^2 + 105105*d^6*e^3*x^3 + 140140*d^5*e^4*x^4 + 126126*d^4*e^5*x^5 + 76440*
d^3*e^6*x^6 + 30030*d^2*e^7*x^7 + 6930*d*e^8*x^8 + 715*e^9*x^9) + 16*a*b^5*x^5*(5005*d^9 + 38610*d^8*e*x + 135
135*d^7*e^2*x^2 + 280280*d^6*e^3*x^3 + 378378*d^5*e^4*x^4 + 343980*d^4*e^5*x^5 + 210210*d^3*e^6*x^6 + 83160*d^
2*e^7*x^7 + 19305*d*e^8*x^8 + 2002*e^9*x^9) + b^6*x^6*(11440*d^9 + 90090*d^8*e*x + 320320*d^7*e^2*x^2 + 672672
*d^6*e^3*x^3 + 917280*d^5*e^4*x^4 + 840840*d^4*e^5*x^5 + 517440*d^3*e^6*x^6 + 205920*d^2*e^7*x^7 + 48048*d*e^8
*x^8 + 5005*e^9*x^9)))/(80080*(a + b*x))
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IntegrateAlgebraic [F] time = 7.95, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^9 \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)^9*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^9*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]
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fricas [B] time = 0.48, size = 892, normalized size = 2.46 \begin {gather*} \frac {1}{16} \, b^{6} e^{9} x^{16} + a^{6} d^{9} x + \frac {1}{5} \, {\left (3 \, b^{6} d e^{8} + 2 \, a b^{5} e^{9}\right )} x^{15} + \frac {3}{14} \, {\left (12 \, b^{6} d^{2} e^{7} + 18 \, a b^{5} d e^{8} + 5 \, a^{2} b^{4} e^{9}\right )} x^{14} + \frac {1}{13} \, {\left (84 \, b^{6} d^{3} e^{6} + 216 \, a b^{5} d^{2} e^{7} + 135 \, a^{2} b^{4} d e^{8} + 20 \, a^{3} b^{3} e^{9}\right )} x^{13} + \frac {1}{4} \, {\left (42 \, b^{6} d^{4} e^{5} + 168 \, a b^{5} d^{3} e^{6} + 180 \, a^{2} b^{4} d^{2} e^{7} + 60 \, a^{3} b^{3} d e^{8} + 5 \, a^{4} b^{2} e^{9}\right )} x^{12} + \frac {3}{11} \, {\left (42 \, b^{6} d^{5} e^{4} + 252 \, a b^{5} d^{4} e^{5} + 420 \, a^{2} b^{4} d^{3} e^{6} + 240 \, a^{3} b^{3} d^{2} e^{7} + 45 \, a^{4} b^{2} d e^{8} + 2 \, a^{5} b e^{9}\right )} x^{11} + \frac {1}{10} \, {\left (84 \, b^{6} d^{6} e^{3} + 756 \, a b^{5} d^{5} e^{4} + 1890 \, a^{2} b^{4} d^{4} e^{5} + 1680 \, a^{3} b^{3} d^{3} e^{6} + 540 \, a^{4} b^{2} d^{2} e^{7} + 54 \, a^{5} b d e^{8} + a^{6} e^{9}\right )} x^{10} + {\left (4 \, b^{6} d^{7} e^{2} + 56 \, a b^{5} d^{6} e^{3} + 210 \, a^{2} b^{4} d^{5} e^{4} + 280 \, a^{3} b^{3} d^{4} e^{5} + 140 \, a^{4} b^{2} d^{3} e^{6} + 24 \, a^{5} b d^{2} e^{7} + a^{6} d e^{8}\right )} x^{9} + \frac {9}{8} \, {\left (b^{6} d^{8} e + 24 \, a b^{5} d^{7} e^{2} + 140 \, a^{2} b^{4} d^{6} e^{3} + 280 \, a^{3} b^{3} d^{5} e^{4} + 210 \, a^{4} b^{2} d^{4} e^{5} + 56 \, a^{5} b d^{3} e^{6} + 4 \, a^{6} d^{2} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{9} + 54 \, a b^{5} d^{8} e + 540 \, a^{2} b^{4} d^{7} e^{2} + 1680 \, a^{3} b^{3} d^{6} e^{3} + 1890 \, a^{4} b^{2} d^{5} e^{4} + 756 \, a^{5} b d^{4} e^{5} + 84 \, a^{6} d^{3} e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{9} + 45 \, a^{2} b^{4} d^{8} e + 240 \, a^{3} b^{3} d^{7} e^{2} + 420 \, a^{4} b^{2} d^{6} e^{3} + 252 \, a^{5} b d^{5} e^{4} + 42 \, a^{6} d^{4} e^{5}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} d^{9} + 60 \, a^{3} b^{3} d^{8} e + 180 \, a^{4} b^{2} d^{7} e^{2} + 168 \, a^{5} b d^{6} e^{3} + 42 \, a^{6} d^{5} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{9} + 135 \, a^{4} b^{2} d^{8} e + 216 \, a^{5} b d^{7} e^{2} + 84 \, a^{6} d^{6} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{9} + 18 \, a^{5} b d^{8} e + 12 \, a^{6} d^{7} e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b d^{9} + 3 \, a^{6} d^{8} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
1/16*b^6*e^9*x^16 + a^6*d^9*x + 1/5*(3*b^6*d*e^8 + 2*a*b^5*e^9)*x^15 + 3/14*(12*b^6*d^2*e^7 + 18*a*b^5*d*e^8 +
5*a^2*b^4*e^9)*x^14 + 1/13*(84*b^6*d^3*e^6 + 216*a*b^5*d^2*e^7 + 135*a^2*b^4*d*e^8 + 20*a^3*b^3*e^9)*x^13 + 1
/4*(42*b^6*d^4*e^5 + 168*a*b^5*d^3*e^6 + 180*a^2*b^4*d^2*e^7 + 60*a^3*b^3*d*e^8 + 5*a^4*b^2*e^9)*x^12 + 3/11*(
42*b^6*d^5*e^4 + 252*a*b^5*d^4*e^5 + 420*a^2*b^4*d^3*e^6 + 240*a^3*b^3*d^2*e^7 + 45*a^4*b^2*d*e^8 + 2*a^5*b*e^
9)*x^11 + 1/10*(84*b^6*d^6*e^3 + 756*a*b^5*d^5*e^4 + 1890*a^2*b^4*d^4*e^5 + 1680*a^3*b^3*d^3*e^6 + 540*a^4*b^2
*d^2*e^7 + 54*a^5*b*d*e^8 + a^6*e^9)*x^10 + (4*b^6*d^7*e^2 + 56*a*b^5*d^6*e^3 + 210*a^2*b^4*d^5*e^4 + 280*a^3*
b^3*d^4*e^5 + 140*a^4*b^2*d^3*e^6 + 24*a^5*b*d^2*e^7 + a^6*d*e^8)*x^9 + 9/8*(b^6*d^8*e + 24*a*b^5*d^7*e^2 + 14
0*a^2*b^4*d^6*e^3 + 280*a^3*b^3*d^5*e^4 + 210*a^4*b^2*d^4*e^5 + 56*a^5*b*d^3*e^6 + 4*a^6*d^2*e^7)*x^8 + 1/7*(b
^6*d^9 + 54*a*b^5*d^8*e + 540*a^2*b^4*d^7*e^2 + 1680*a^3*b^3*d^6*e^3 + 1890*a^4*b^2*d^5*e^4 + 756*a^5*b*d^4*e^
5 + 84*a^6*d^3*e^6)*x^7 + 1/2*(2*a*b^5*d^9 + 45*a^2*b^4*d^8*e + 240*a^3*b^3*d^7*e^2 + 420*a^4*b^2*d^6*e^3 + 25
2*a^5*b*d^5*e^4 + 42*a^6*d^4*e^5)*x^6 + 3/5*(5*a^2*b^4*d^9 + 60*a^3*b^3*d^8*e + 180*a^4*b^2*d^7*e^2 + 168*a^5*
b*d^6*e^3 + 42*a^6*d^5*e^4)*x^5 + 1/4*(20*a^3*b^3*d^9 + 135*a^4*b^2*d^8*e + 216*a^5*b*d^7*e^2 + 84*a^6*d^6*e^3
)*x^4 + (5*a^4*b^2*d^9 + 18*a^5*b*d^8*e + 12*a^6*d^7*e^2)*x^3 + 3/2*(2*a^5*b*d^9 + 3*a^6*d^8*e)*x^2
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giac [B] time = 0.29, size = 1387, normalized size = 3.83
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
1/16*b^6*x^16*e^9*sgn(b*x + a) + 3/5*b^6*d*x^15*e^8*sgn(b*x + a) + 18/7*b^6*d^2*x^14*e^7*sgn(b*x + a) + 84/13*
b^6*d^3*x^13*e^6*sgn(b*x + a) + 21/2*b^6*d^4*x^12*e^5*sgn(b*x + a) + 126/11*b^6*d^5*x^11*e^4*sgn(b*x + a) + 42
/5*b^6*d^6*x^10*e^3*sgn(b*x + a) + 4*b^6*d^7*x^9*e^2*sgn(b*x + a) + 9/8*b^6*d^8*x^8*e*sgn(b*x + a) + 1/7*b^6*d
^9*x^7*sgn(b*x + a) + 2/5*a*b^5*x^15*e^9*sgn(b*x + a) + 27/7*a*b^5*d*x^14*e^8*sgn(b*x + a) + 216/13*a*b^5*d^2*
x^13*e^7*sgn(b*x + a) + 42*a*b^5*d^3*x^12*e^6*sgn(b*x + a) + 756/11*a*b^5*d^4*x^11*e^5*sgn(b*x + a) + 378/5*a*
b^5*d^5*x^10*e^4*sgn(b*x + a) + 56*a*b^5*d^6*x^9*e^3*sgn(b*x + a) + 27*a*b^5*d^7*x^8*e^2*sgn(b*x + a) + 54/7*a
*b^5*d^8*x^7*e*sgn(b*x + a) + a*b^5*d^9*x^6*sgn(b*x + a) + 15/14*a^2*b^4*x^14*e^9*sgn(b*x + a) + 135/13*a^2*b^
4*d*x^13*e^8*sgn(b*x + a) + 45*a^2*b^4*d^2*x^12*e^7*sgn(b*x + a) + 1260/11*a^2*b^4*d^3*x^11*e^6*sgn(b*x + a) +
189*a^2*b^4*d^4*x^10*e^5*sgn(b*x + a) + 210*a^2*b^4*d^5*x^9*e^4*sgn(b*x + a) + 315/2*a^2*b^4*d^6*x^8*e^3*sgn(
b*x + a) + 540/7*a^2*b^4*d^7*x^7*e^2*sgn(b*x + a) + 45/2*a^2*b^4*d^8*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^9*x^5*sg
n(b*x + a) + 20/13*a^3*b^3*x^13*e^9*sgn(b*x + a) + 15*a^3*b^3*d*x^12*e^8*sgn(b*x + a) + 720/11*a^3*b^3*d^2*x^1
1*e^7*sgn(b*x + a) + 168*a^3*b^3*d^3*x^10*e^6*sgn(b*x + a) + 280*a^3*b^3*d^4*x^9*e^5*sgn(b*x + a) + 315*a^3*b^
3*d^5*x^8*e^4*sgn(b*x + a) + 240*a^3*b^3*d^6*x^7*e^3*sgn(b*x + a) + 120*a^3*b^3*d^7*x^6*e^2*sgn(b*x + a) + 36*
a^3*b^3*d^8*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^9*x^4*sgn(b*x + a) + 5/4*a^4*b^2*x^12*e^9*sgn(b*x + a) + 135/11*a
^4*b^2*d*x^11*e^8*sgn(b*x + a) + 54*a^4*b^2*d^2*x^10*e^7*sgn(b*x + a) + 140*a^4*b^2*d^3*x^9*e^6*sgn(b*x + a) +
945/4*a^4*b^2*d^4*x^8*e^5*sgn(b*x + a) + 270*a^4*b^2*d^5*x^7*e^4*sgn(b*x + a) + 210*a^4*b^2*d^6*x^6*e^3*sgn(b
*x + a) + 108*a^4*b^2*d^7*x^5*e^2*sgn(b*x + a) + 135/4*a^4*b^2*d^8*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^9*x^3*sgn(
b*x + a) + 6/11*a^5*b*x^11*e^9*sgn(b*x + a) + 27/5*a^5*b*d*x^10*e^8*sgn(b*x + a) + 24*a^5*b*d^2*x^9*e^7*sgn(b*
x + a) + 63*a^5*b*d^3*x^8*e^6*sgn(b*x + a) + 108*a^5*b*d^4*x^7*e^5*sgn(b*x + a) + 126*a^5*b*d^5*x^6*e^4*sgn(b*
x + a) + 504/5*a^5*b*d^6*x^5*e^3*sgn(b*x + a) + 54*a^5*b*d^7*x^4*e^2*sgn(b*x + a) + 18*a^5*b*d^8*x^3*e*sgn(b*x
+ a) + 3*a^5*b*d^9*x^2*sgn(b*x + a) + 1/10*a^6*x^10*e^9*sgn(b*x + a) + a^6*d*x^9*e^8*sgn(b*x + a) + 9/2*a^6*d
^2*x^8*e^7*sgn(b*x + a) + 12*a^6*d^3*x^7*e^6*sgn(b*x + a) + 21*a^6*d^4*x^6*e^5*sgn(b*x + a) + 126/5*a^6*d^5*x^
5*e^4*sgn(b*x + a) + 21*a^6*d^6*x^4*e^3*sgn(b*x + a) + 12*a^6*d^7*x^3*e^2*sgn(b*x + a) + 9/2*a^6*d^8*x^2*e*sgn
(b*x + a) + a^6*d^9*x*sgn(b*x + a)
________________________________________________________________________________________
maple [B] time = 0.05, size = 1034, normalized size = 2.86 \begin {gather*} \frac {\left (5005 b^{6} e^{9} x^{15}+32032 x^{14} a \,b^{5} e^{9}+48048 x^{14} b^{6} d \,e^{8}+85800 x^{13} a^{2} b^{4} e^{9}+308880 x^{13} a \,b^{5} d \,e^{8}+205920 x^{13} b^{6} d^{2} e^{7}+123200 x^{12} a^{3} b^{3} e^{9}+831600 x^{12} a^{2} b^{4} d \,e^{8}+1330560 x^{12} a \,b^{5} d^{2} e^{7}+517440 x^{12} b^{6} d^{3} e^{6}+100100 x^{11} a^{4} b^{2} e^{9}+1201200 x^{11} a^{3} b^{3} d \,e^{8}+3603600 x^{11} a^{2} b^{4} d^{2} e^{7}+3363360 x^{11} a \,b^{5} d^{3} e^{6}+840840 x^{11} b^{6} d^{4} e^{5}+43680 x^{10} a^{5} b \,e^{9}+982800 x^{10} a^{4} b^{2} d \,e^{8}+5241600 x^{10} a^{3} b^{3} d^{2} e^{7}+9172800 x^{10} a^{2} b^{4} d^{3} e^{6}+5503680 x^{10} a \,b^{5} d^{4} e^{5}+917280 x^{10} b^{6} d^{5} e^{4}+8008 x^{9} a^{6} e^{9}+432432 x^{9} a^{5} b d \,e^{8}+4324320 x^{9} a^{4} b^{2} d^{2} e^{7}+13453440 x^{9} a^{3} b^{3} d^{3} e^{6}+15135120 x^{9} a^{2} b^{4} d^{4} e^{5}+6054048 x^{9} a \,b^{5} d^{5} e^{4}+672672 x^{9} b^{6} d^{6} e^{3}+80080 a^{6} d \,e^{8} x^{8}+1921920 a^{5} b \,d^{2} e^{7} x^{8}+11211200 a^{4} b^{2} d^{3} e^{6} x^{8}+22422400 a^{3} b^{3} d^{4} e^{5} x^{8}+16816800 a^{2} b^{4} d^{5} e^{4} x^{8}+4484480 a \,b^{5} d^{6} e^{3} x^{8}+320320 b^{6} d^{7} e^{2} x^{8}+360360 x^{7} a^{6} d^{2} e^{7}+5045040 x^{7} a^{5} b \,d^{3} e^{6}+18918900 x^{7} a^{4} b^{2} d^{4} e^{5}+25225200 x^{7} a^{3} b^{3} d^{5} e^{4}+12612600 x^{7} a^{2} b^{4} d^{6} e^{3}+2162160 x^{7} a \,b^{5} d^{7} e^{2}+90090 x^{7} b^{6} d^{8} e +960960 x^{6} a^{6} d^{3} e^{6}+8648640 x^{6} a^{5} b \,d^{4} e^{5}+21621600 x^{6} a^{4} b^{2} d^{5} e^{4}+19219200 x^{6} a^{3} b^{3} d^{6} e^{3}+6177600 x^{6} a^{2} b^{4} d^{7} e^{2}+617760 x^{6} a \,b^{5} d^{8} e +11440 x^{6} b^{6} d^{9}+1681680 x^{5} a^{6} d^{4} e^{5}+10090080 x^{5} a^{5} b \,d^{5} e^{4}+16816800 x^{5} a^{4} b^{2} d^{6} e^{3}+9609600 x^{5} a^{3} b^{3} d^{7} e^{2}+1801800 x^{5} a^{2} b^{4} d^{8} e +80080 x^{5} a \,b^{5} d^{9}+2018016 x^{4} a^{6} d^{5} e^{4}+8072064 x^{4} a^{5} b \,d^{6} e^{3}+8648640 x^{4} a^{4} b^{2} d^{7} e^{2}+2882880 x^{4} a^{3} b^{3} d^{8} e +240240 x^{4} a^{2} b^{4} d^{9}+1681680 x^{3} a^{6} d^{6} e^{3}+4324320 x^{3} a^{5} b \,d^{7} e^{2}+2702700 x^{3} a^{4} b^{2} d^{8} e +400400 x^{3} a^{3} b^{3} d^{9}+960960 a^{6} d^{7} e^{2} x^{2}+1441440 a^{5} b \,d^{8} e \,x^{2}+400400 a^{4} b^{2} d^{9} x^{2}+360360 x \,a^{6} d^{8} e +240240 x \,a^{5} b \,d^{9}+80080 a^{6} d^{9}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{80080 \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)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
1/80080*x*(5005*b^6*e^9*x^15+32032*a*b^5*e^9*x^14+48048*b^6*d*e^8*x^14+85800*a^2*b^4*e^9*x^13+308880*a*b^5*d*e
^8*x^13+205920*b^6*d^2*e^7*x^13+123200*a^3*b^3*e^9*x^12+831600*a^2*b^4*d*e^8*x^12+1330560*a*b^5*d^2*e^7*x^12+5
17440*b^6*d^3*e^6*x^12+100100*a^4*b^2*e^9*x^11+1201200*a^3*b^3*d*e^8*x^11+3603600*a^2*b^4*d^2*e^7*x^11+3363360
*a*b^5*d^3*e^6*x^11+840840*b^6*d^4*e^5*x^11+43680*a^5*b*e^9*x^10+982800*a^4*b^2*d*e^8*x^10+5241600*a^3*b^3*d^2
*e^7*x^10+9172800*a^2*b^4*d^3*e^6*x^10+5503680*a*b^5*d^4*e^5*x^10+917280*b^6*d^5*e^4*x^10+8008*a^6*e^9*x^9+432
432*a^5*b*d*e^8*x^9+4324320*a^4*b^2*d^2*e^7*x^9+13453440*a^3*b^3*d^3*e^6*x^9+15135120*a^2*b^4*d^4*e^5*x^9+6054
048*a*b^5*d^5*e^4*x^9+672672*b^6*d^6*e^3*x^9+80080*a^6*d*e^8*x^8+1921920*a^5*b*d^2*e^7*x^8+11211200*a^4*b^2*d^
3*e^6*x^8+22422400*a^3*b^3*d^4*e^5*x^8+16816800*a^2*b^4*d^5*e^4*x^8+4484480*a*b^5*d^6*e^3*x^8+320320*b^6*d^7*e
^2*x^8+360360*a^6*d^2*e^7*x^7+5045040*a^5*b*d^3*e^6*x^7+18918900*a^4*b^2*d^4*e^5*x^7+25225200*a^3*b^3*d^5*e^4*
x^7+12612600*a^2*b^4*d^6*e^3*x^7+2162160*a*b^5*d^7*e^2*x^7+90090*b^6*d^8*e*x^7+960960*a^6*d^3*e^6*x^6+8648640*
a^5*b*d^4*e^5*x^6+21621600*a^4*b^2*d^5*e^4*x^6+19219200*a^3*b^3*d^6*e^3*x^6+6177600*a^2*b^4*d^7*e^2*x^6+617760
*a*b^5*d^8*e*x^6+11440*b^6*d^9*x^6+1681680*a^6*d^4*e^5*x^5+10090080*a^5*b*d^5*e^4*x^5+16816800*a^4*b^2*d^6*e^3
*x^5+9609600*a^3*b^3*d^7*e^2*x^5+1801800*a^2*b^4*d^8*e*x^5+80080*a*b^5*d^9*x^5+2018016*a^6*d^5*e^4*x^4+8072064
*a^5*b*d^6*e^3*x^4+8648640*a^4*b^2*d^7*e^2*x^4+2882880*a^3*b^3*d^8*e*x^4+240240*a^2*b^4*d^9*x^4+1681680*a^6*d^
6*e^3*x^3+4324320*a^5*b*d^7*e^2*x^3+2702700*a^4*b^2*d^8*e*x^3+400400*a^3*b^3*d^9*x^3+960960*a^6*d^7*e^2*x^2+14
41440*a^5*b*d^8*e*x^2+400400*a^4*b^2*d^9*x^2+360360*a^6*d^8*e*x+240240*a^5*b*d^9*x+80080*a^6*d^9)*((b*x+a)^2)^
(5/2)/(b*x+a)^5
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maxima [B] time = 0.80, size = 3175, normalized size = 8.77
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
1/16*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^9*x^9/b - 5/48*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^9*x^8/b^2 + 11/84*(b
^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^9*x^7/b^3 - 23/156*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^9*x^6/b^4 + 49/31
2*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^9*x^5/b^5 - 557/3432*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*e^9*x^4/b^6 +
283/1716*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6*e^9*x^3/b^7 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^9*x + 1/6*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^10*e^9*x/b^9 - 95/572*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7*e^9*x^2/b^8 + 1/6*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^9/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^11*e^9/b^10 + 381/2288*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*a^8*e^9*x/b^9 - 2669/16016*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^9*e^9/b^10 + 1/15*(9*b
*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^8/b^2 - 23/210*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)
^(7/2)*a*x^7/b^3 + 9/14*(4*b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^7/b^2 + 53/390*(9*b*d*e^8 +
a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^6/b^4 - 27/26*(4*b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(
7/2)*a*x^6/b^3 + 12/13*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^6/b^2 - 59/390*(9*b*d*e^8
+ a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^5/b^5 + 33/26*(4*b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)*a^2*x^5/b^4 - 19/13*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^5/b^3 + 7/2*(3*b*d
^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^5/b^2 + 137/858*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x
+ a^2)^(7/2)*a^4*x^4/b^6 - 399/286*(4*b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^4/b^5 + 251/
143*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^4/b^4 - 119/22*(3*b*d^4*e^5 + 2*a*d^3*e^
6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^4/b^3 + 126/11*(b*d^5*e^4 + a*d^4*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*
x^4/b^2 - 703/4290*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*x^3/b^7 + 417/286*(4*b*d^2*e^7 + a*
d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x^3/b^6 - 272/143*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*a^3*x^3/b^5 + 70/11*(3*b*d^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^3/b^4 - 189/1
1*(b*d^5*e^4 + a*d^4*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 21/5*(2*b*d^6*e^3 + 3*a*d^5*e^4)*(b^2*x^
2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 - 1/6*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9*x/b^9 + 3/2*(4*
b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^8*x/b^8 - 2*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*
b*x + a^2)^(5/2)*a^7*x/b^7 + 7*(3*b*d^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*x/b^6 - 21*(b*d
^5*e^4 + a*d^4*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*x/b^5 + 7*(2*b*d^6*e^3 + 3*a*d^5*e^4)*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*a^4*x/b^4 - 2*(3*b*d^7*e^2 + 7*a*d^6*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 3/2*(b*d
^8*e + 4*a*d^7*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(b*d^9 + 9*a*d^8*e)*(b^2*x^2 + 2*a*b*x + a
^2)^(5/2)*a*x/b + 237/1430*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6*x^2/b^8 - 425/286*(4*b*d^2*
e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*x^2/b^7 + 844/429*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*a^4*x^2/b^6 - 224/33*(3*b*d^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^2/b^
5 + 217/11*(b*d^5*e^4 + a*d^4*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^2/b^4 - 91/15*(2*b*d^6*e^3 + 3*a*d^5*
e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^2/b^3 + 4/3*(3*b*d^7*e^2 + 7*a*d^6*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/
2)*x^2/b^2 - 1/6*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^10/b^10 + 3/2*(4*b*d^2*e^7 + a*d*e^8)*(
b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9/b^9 - 2*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^8/b^8
+ 7*(3*b*d^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7/b^7 - 21*(b*d^5*e^4 + a*d^4*e^5)*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*a^6/b^6 + 7*(2*b*d^6*e^3 + 3*a*d^5*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^5 - 2*(3
*b*d^7*e^2 + 7*a*d^6*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 3/2*(b*d^8*e + 4*a*d^7*e^2)*(b^2*x^2 + 2*a
*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(b*d^9 + 9*a*d^8*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 - 119/715*(9*b*d*e
^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7*x/b^9 + 214/143*(4*b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a
^2)^(7/2)*a^6*x/b^8 - 1709/858*(7*b*d^3*e^6 + 3*a*d^2*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*x/b^7 + 917/132
*(3*b*d^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x/b^6 - 455/22*(b*d^5*e^4 + a*d^4*e^5)*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*a^3*x/b^5 + 203/30*(2*b*d^6*e^3 + 3*a*d^5*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x
/b^4 - 11/6*(3*b*d^7*e^2 + 7*a*d^6*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 9/8*(b*d^8*e + 4*a*d^7*e^2)*
(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 + 834/5005*(9*b*d*e^8 + a*e^9)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^8/b^10
- 1501/1001*(4*b*d^2*e^7 + a*d*e^8)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7/b^9 + 1715/858*(7*b*d^3*e^6 + 3*a*d^2*
e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6/b^8 - 923/132*(3*b*d^4*e^5 + 2*a*d^3*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(
7/2)*a^5/b^7 + 461/22*(b*d^5*e^4 + a*d^4*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4/b^6 - 209/30*(2*b*d^6*e^3 +
3*a*d^5*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 83/42*(3*b*d^7*e^2 + 7*a*d^6*e^3)*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*a^2/b^4 - 81/56*(b*d^8*e + 4*a*d^7*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(b*d^9 + 9*a*d^
8*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 )}^9\,{\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)^9*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
[Out]
int((a + b*x)*(d + e*x)^9*(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 )^{9} \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)**9*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Integral((a + b*x)*(d + e*x)**9*((a + b*x)**2)**(5/2), x)
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