3.2059 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=216 \[ -\frac{14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac{42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac{14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac{70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac{42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac{14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac{2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac{2 b^7 (d+e x)^{21/2}}{21 e^8} \]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(7/2))/(7*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^8) - (42*b^2*(b*d - a*e
)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(b*d - a*e)^4*(d + e*x)^(13/2))/(13*e^8) - (14*b^4*(b*d - a*e)^3*(d +
e*x)^(15/2))/(3*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(17/2))/(17*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(19/2
))/(19*e^8) + (2*b^7*(d + e*x)^(21/2))/(21*e^8)
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Rubi [A] time = 0.0758327, antiderivative size = 216, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used =
{27, 43} \[ -\frac{14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac{42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac{14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac{70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac{42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac{14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac{2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac{2 b^7 (d+e x)^{21/2}}{21 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(7/2))/(7*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^8) - (42*b^2*(b*d - a*e
)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(b*d - a*e)^4*(d + e*x)^(13/2))/(13*e^8) - (14*b^4*(b*d - a*e)^3*(d +
e*x)^(15/2))/(3*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(17/2))/(17*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(19/2
))/(19*e^8) + (2*b^7*(d + e*x)^(21/2))/(21*e^8)
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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])
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^7 (d+e x)^{5/2}}{e^7}+\frac{7 b (b d-a e)^6 (d+e x)^{7/2}}{e^7}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{9/2}}{e^7}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{11/2}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{13/2}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{15/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{17/2}}{e^7}+\frac{b^7 (d+e x)^{19/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^7 (d+e x)^{7/2}}{7 e^8}+\frac{14 b (b d-a e)^6 (d+e x)^{9/2}}{9 e^8}-\frac{42 b^2 (b d-a e)^5 (d+e x)^{11/2}}{11 e^8}+\frac{70 b^3 (b d-a e)^4 (d+e x)^{13/2}}{13 e^8}-\frac{14 b^4 (b d-a e)^3 (d+e x)^{15/2}}{3 e^8}+\frac{42 b^5 (b d-a e)^2 (d+e x)^{17/2}}{17 e^8}-\frac{14 b^6 (b d-a e) (d+e x)^{19/2}}{19 e^8}+\frac{2 b^7 (d+e x)^{21/2}}{21 e^8}\\ \end{align*}
Mathematica [A] time = 0.153568, size = 167, normalized size = 0.77 \[ \frac{2 (d+e x)^{7/2} \left (-5555277 b^2 (d+e x)^2 (b d-a e)^5+7834365 b^3 (d+e x)^3 (b d-a e)^4-6789783 b^4 (d+e x)^4 (b d-a e)^3+3594591 b^5 (d+e x)^5 (b d-a e)^2-1072071 b^6 (d+e x)^6 (b d-a e)+2263261 b (d+e x) (b d-a e)^6-415701 (b d-a e)^7+138567 b^7 (d+e x)^7\right )}{2909907 e^8} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(-415701*(b*d - a*e)^7 + 2263261*b*(b*d - a*e)^6*(d + e*x) - 5555277*b^2*(b*d - a*e)^5*(d +
e*x)^2 + 7834365*b^3*(b*d - a*e)^4*(d + e*x)^3 - 6789783*b^4*(b*d - a*e)^3*(d + e*x)^4 + 3594591*b^5*(b*d - a
*e)^2*(d + e*x)^5 - 1072071*b^6*(b*d - a*e)*(d + e*x)^6 + 138567*b^7*(d + e*x)^7))/(2909907*e^8)
________________________________________________________________________________________
Maple [B] time = 0.007, size = 498, normalized size = 2.3 \begin{align*}{\frac{277134\,{b}^{7}{x}^{7}{e}^{7}+2144142\,a{b}^{6}{e}^{7}{x}^{6}-204204\,{b}^{7}d{e}^{6}{x}^{6}+7189182\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}-1513512\,a{b}^{6}d{e}^{6}{x}^{5}+144144\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}+13579566\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}-4792788\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}+1009008\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}-96096\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}+15668730\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}-8356656\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}+2949408\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}-620928\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}+59136\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+11110554\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-8546580\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+4558176\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-1608768\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+338688\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-32256\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+4526522\,{a}^{6}b{e}^{7}x-4938024\,{a}^{5}{b}^{2}d{e}^{6}x+3798480\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-2025856\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+715008\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-150528\,a{b}^{6}{d}^{5}{e}^{2}x+14336\,{b}^{7}{d}^{6}ex+831402\,{a}^{7}{e}^{7}-1293292\,{a}^{6}bd{e}^{6}+1410864\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-1085280\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+578816\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-204288\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+43008\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{2909907\,{e}^{8}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/2909907*(e*x+d)^(7/2)*(138567*b^7*e^7*x^7+1072071*a*b^6*e^7*x^6-102102*b^7*d*e^6*x^6+3594591*a^2*b^5*e^7*x^5
-756756*a*b^6*d*e^6*x^5+72072*b^7*d^2*e^5*x^5+6789783*a^3*b^4*e^7*x^4-2396394*a^2*b^5*d*e^6*x^4+504504*a*b^6*d
^2*e^5*x^4-48048*b^7*d^3*e^4*x^4+7834365*a^4*b^3*e^7*x^3-4178328*a^3*b^4*d*e^6*x^3+1474704*a^2*b^5*d^2*e^5*x^3
-310464*a*b^6*d^3*e^4*x^3+29568*b^7*d^4*e^3*x^3+5555277*a^5*b^2*e^7*x^2-4273290*a^4*b^3*d*e^6*x^2+2279088*a^3*
b^4*d^2*e^5*x^2-804384*a^2*b^5*d^3*e^4*x^2+169344*a*b^6*d^4*e^3*x^2-16128*b^7*d^5*e^2*x^2+2263261*a^6*b*e^7*x-
2469012*a^5*b^2*d*e^6*x+1899240*a^4*b^3*d^2*e^5*x-1012928*a^3*b^4*d^3*e^4*x+357504*a^2*b^5*d^4*e^3*x-75264*a*b
^6*d^5*e^2*x+7168*b^7*d^6*e*x+415701*a^7*e^7-646646*a^6*b*d*e^6+705432*a^5*b^2*d^2*e^5-542640*a^4*b^3*d^3*e^4+
289408*a^3*b^4*d^4*e^3-102144*a^2*b^5*d^5*e^2+21504*a*b^6*d^6*e-2048*b^7*d^7)/e^8
________________________________________________________________________________________
Maxima [B] time = 1.01747, size = 616, normalized size = 2.85 \begin{align*} \frac{2 \,{\left (138567 \,{\left (e x + d\right )}^{\frac{21}{2}} b^{7} - 1072071 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{19}{2}} + 3594591 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{17}{2}} - 6789783 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 7834365 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 5555277 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 2263261 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 415701 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{2909907 \, e^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/2909907*(138567*(e*x + d)^(21/2)*b^7 - 1072071*(b^7*d - a*b^6*e)*(e*x + d)^(19/2) + 3594591*(b^7*d^2 - 2*a*b
^6*d*e + a^2*b^5*e^2)*(e*x + d)^(17/2) - 6789783*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*
x + d)^(15/2) + 7834365*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*(e*x + d
)^(13/2) - 5555277*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*
b^2*e^5)*(e*x + d)^(11/2) + 2263261*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^
4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d)^(9/2) - 415701*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^
5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d)^(7/2
))/e^8
________________________________________________________________________________________
Fricas [B] time = 1.36445, size = 1882, normalized size = 8.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/2909907*(138567*b^7*e^10*x^10 - 2048*b^7*d^10 + 21504*a*b^6*d^9*e - 102144*a^2*b^5*d^8*e^2 + 289408*a^3*b^4*
d^7*e^3 - 542640*a^4*b^3*d^6*e^4 + 705432*a^5*b^2*d^5*e^5 - 646646*a^6*b*d^4*e^6 + 415701*a^7*d^3*e^7 + 7293*(
43*b^7*d*e^9 + 147*a*b^6*e^10)*x^9 + 3861*(47*b^7*d^2*e^8 + 637*a*b^6*d*e^9 + 931*a^2*b^5*e^10)*x^8 + 429*(b^7
*d^3*e^7 + 3381*a*b^6*d^2*e^8 + 19551*a^2*b^5*d*e^9 + 15827*a^3*b^4*e^10)*x^7 - 231*(2*b^7*d^4*e^6 - 21*a*b^6*
d^3*e^7 - 21945*a^2*b^5*d^2*e^8 - 70091*a^3*b^4*d*e^9 - 33915*a^4*b^3*e^10)*x^6 + 63*(8*b^7*d^5*e^5 - 84*a*b^6
*d^4*e^6 + 399*a^2*b^5*d^3*e^7 + 160531*a^3*b^4*d^2*e^8 + 305235*a^4*b^3*d*e^9 + 88179*a^5*b^2*e^10)*x^5 - 7*(
80*b^7*d^6*e^4 - 840*a*b^6*d^5*e^5 + 3990*a^2*b^5*d^4*e^6 - 11305*a^3*b^4*d^3*e^7 - 1797495*a^4*b^3*d^2*e^8 -
2028117*a^5*b^2*d*e^9 - 323323*a^6*b*e^10)*x^4 + (640*b^7*d^7*e^3 - 6720*a*b^6*d^6*e^4 + 31920*a^2*b^5*d^5*e^5
- 90440*a^3*b^4*d^4*e^6 + 169575*a^4*b^3*d^3*e^7 + 9964227*a^5*b^2*d^2*e^8 + 6143137*a^6*b*d*e^9 + 415701*a^7
*e^10)*x^3 - 3*(256*b^7*d^8*e^2 - 2688*a*b^6*d^7*e^3 + 12768*a^2*b^5*d^6*e^4 - 36176*a^3*b^4*d^5*e^5 + 67830*a
^4*b^3*d^4*e^6 - 88179*a^5*b^2*d^3*e^7 - 1616615*a^6*b*d^2*e^8 - 415701*a^7*d*e^9)*x^2 + (1024*b^7*d^9*e - 107
52*a*b^6*d^8*e^2 + 51072*a^2*b^5*d^7*e^3 - 144704*a^3*b^4*d^6*e^4 + 271320*a^4*b^3*d^5*e^5 - 352716*a^5*b^2*d^
4*e^6 + 323323*a^6*b*d^3*e^7 + 1247103*a^7*d^2*e^8)*x)*sqrt(e*x + d)/e^8
________________________________________________________________________________________
Sympy [A] time = 58.0706, size = 2096, normalized size = 9.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
a**7*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**7*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**7*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
14*a**6*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 28*a**6*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*
(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 14*a**6*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/
5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 42*a**5*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 84*a**5*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/
5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 42*a**5*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*
x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 70*a**4*b**3*d
**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4
+ 140*a**4*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d +
e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 70*a**4*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) -
10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
4 + 70*a**3*b**4*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(
d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 140*a**3*b**4*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(
5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/
13)/e**5 + 70*a**3*b**4*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20
*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**
5 + 42*a**2*b**5*d**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2
*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 84*a**2*b**5*d*(d**6*(d + e*x)**
(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e
*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 42*a**2*b**5*(-d**7*(d + e*x)**(3/2)/
3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/
2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 14*a*b**6*d**2*
(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9
+ 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 28*a*b**6*d*(-d**7*(
d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3
*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 +
14*a*b**6*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)*
*(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d
+ e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7 + 2*b**7*d**2*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5
/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x
)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**8 + 4*b**7*d*(d**8*(d + e*x)**(3/2)/3 - 8*d
**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 -
56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19
)/e**8 + 2*b**7*(-d**9*(d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(
d + e*x)**(9/2)/3 - 126*d**5*(d + e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/
5 + 36*d**2*(d + e*x)**(17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**8
________________________________________________________________________________________
Giac [B] time = 1.32812, size = 2511, normalized size = 11.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/14549535*(6789783*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*b*d^2*e^(-1) + 2909907*(15*(x*e + d)^(7/2) -
42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b^2*d^2*e^(-2) + 1616615*(35*(x*e + d)^(9/2) - 135*(x*e +
d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^4*b^3*d^2*e^(-3) + 146965*(315*(x*e + d)^(11
/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)
*a^3*b^4*d^2*e^(-4) + 33915*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 1287
0*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^2*b^5*d^2*e^(-5) + 2261*(3003*(
x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(
x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a*b^6*d^2*e^(-6) + 133*(6435*(x*e
+ d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x
*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*b^7*d
^2*e^(-7) + 4849845*(x*e + d)^(3/2)*a^7*d^2 + 1939938*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d
)^(3/2)*d^2)*a^6*b*d*e^(-1) + 1939938*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 -
105*(x*e + d)^(3/2)*d^3)*a^5*b^2*d*e^(-2) + 293930*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e
+ d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^4*b^3*d*e^(-3) + 113050*(693*(x*e + d)
^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/
2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^3*b^4*d*e^(-4) + 13566*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d
+ 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*
d^5 + 15015*(x*e + d)^(3/2)*d^6)*a^2*b^5*d*e^(-5) + 1862*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 1
76715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)
*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*a*b^6*d*e^(-6) + 14*(109395*(x*e + d)^(19/2) -
978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(
11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*
(x*e + d)^(3/2)*d^8)*b^7*d*e^(-7) + 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^7*d + 323323*(35*(x*e
+ d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^6*b*e^(-1) + 88179*(
315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*
e + d)^(3/2)*d^4)*a^5*b^2*e^(-2) + 56525*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/
2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^4*b^3*e^(-3) + 113
05*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3
+ 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a^3*b^4*e^(-4) + 2793*(6
435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 +
425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d
^7)*a^2*b^5*e^(-5) + 49*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 -
8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d
)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*a*b^6*e^(-6) + 3*(230945*(x*e + d)^(21
/2) - 2297295*(x*e + d)^(19/2)*d + 10270260*(x*e + d)^(17/2)*d^2 - 27159132*(x*e + d)^(15/2)*d^3 + 47006190*(x
*e + d)^(13/2)*d^4 - 55552770*(x*e + d)^(11/2)*d^5 + 45265220*(x*e + d)^(9/2)*d^6 - 24942060*(x*e + d)^(7/2)*d
^7 + 8729721*(x*e + d)^(5/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)*b^7*e^(-7) + 138567*(15*(x*e + d)^(7/2) - 42*(
x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^7)*e^(-1)