3.22.10 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [2110]

Optimal. Leaf size=374 \[ \frac {2 (b d-a e)^6 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {4 b (b d-a e)^5 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^4 (b d-a e)^2 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 e^7 (a+b x)} \]

[Out]

2/7*(-a*e+b*d)^6*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-4/3*b*(-a*e+b*d)^5*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2
)/e^7/(b*x+a)+30/11*b^2*(-a*e+b*d)^4*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-40/13*b^3*(-a*e+b*d)^3*(e*x+
d)^(13/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2*b^4*(-a*e+b*d)^2*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-12/17*
b^5*(-a*e+b*d)*(e*x+d)^(17/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/19*b^6*(e*x+d)^(19/2)*((b*x+a)^2)^(1/2)/e^7/(b*x
+a)

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Rubi [A]
time = 0.12, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 45} \begin {gather*} \frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^4}{11 e^7 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^5}{3 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^6}{7 e^7 (a+b x)}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (b d-a e)}{17 e^7 (a+b x)}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)^2}{e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^3}{13 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (4*b*(b*d - a*e)^5*(d + e*
x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(11/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(13*e^7*(a + b*x)) + (2*b^4*(b*d - a*e)^2*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (
12*b^5*(b*d - a*e)*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(19/2
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*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 45

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 784

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)^{5/2} \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)^{5/2} \, 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)^{5/2} \, 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)^{5/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{7/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{9/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{15/2}}{e^6}+\frac {b^6 (d+e x)^{17/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^6 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {4 b (b d-a e)^5 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^4 (b d-a e)^2 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 309, normalized size = 0.83 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (138567 a^6 e^6+92378 a^5 b e^5 (-2 d+7 e x)+20995 a^4 b^2 e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+6460 a^3 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 a^2 b^4 e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+38 a b^5 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{969969 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(138567*a^6*e^6 + 92378*a^5*b*e^5*(-2*d + 7*e*x) + 20995*a^4*b^2*e^4*(8*d
^2 - 28*d*e*x + 63*e^2*x^2) + 6460*a^3*b^3*e^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 323*a^2*
b^4*e^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 38*a*b^5*e*(-256*d^5 + 89
6*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + b^6*(1024*d^6 - 3584*d^5*e*
x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6)))/(969969*e^7*
(a + b*x))

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Maple [A]
time = 0.04, size = 393, normalized size = 1.05

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (51051 b^{6} e^{6} x^{6}+342342 a \,b^{5} e^{6} x^{5}-36036 b^{6} d \,e^{5} x^{5}+969969 a^{2} b^{4} e^{6} x^{4}-228228 a \,b^{5} d \,e^{5} x^{4}+24024 b^{6} d^{2} e^{4} x^{4}+1492260 a^{3} b^{3} e^{6} x^{3}-596904 a^{2} b^{4} d \,e^{5} x^{3}+140448 a \,b^{5} d^{2} e^{4} x^{3}-14784 b^{6} d^{3} e^{3} x^{3}+1322685 a^{4} b^{2} e^{6} x^{2}-813960 a^{3} b^{3} d \,e^{5} x^{2}+325584 a^{2} b^{4} d^{2} e^{4} x^{2}-76608 a \,b^{5} d^{3} e^{3} x^{2}+8064 b^{6} d^{4} e^{2} x^{2}+646646 a^{5} b \,e^{6} x -587860 a^{4} b^{2} d \,e^{5} x +361760 a^{3} b^{3} d^{2} e^{4} x -144704 a^{2} b^{4} d^{3} e^{3} x +34048 a \,b^{5} d^{4} e^{2} x -3584 b^{6} d^{5} e x +138567 e^{6} a^{6}-184756 d \,e^{5} a^{5} b +167960 d^{2} e^{4} a^{4} b^{2}-103360 d^{3} e^{3} a^{3} b^{3}+41344 d^{4} e^{2} a^{2} b^{4}-9728 d^{5} e a \,b^{5}+1024 d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{969969 e^{7} \left (b x +a \right )^{5}}\)	\(393\)
default	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (51051 b^{6} e^{6} x^{6}+342342 a \,b^{5} e^{6} x^{5}-36036 b^{6} d \,e^{5} x^{5}+969969 a^{2} b^{4} e^{6} x^{4}-228228 a \,b^{5} d \,e^{5} x^{4}+24024 b^{6} d^{2} e^{4} x^{4}+1492260 a^{3} b^{3} e^{6} x^{3}-596904 a^{2} b^{4} d \,e^{5} x^{3}+140448 a \,b^{5} d^{2} e^{4} x^{3}-14784 b^{6} d^{3} e^{3} x^{3}+1322685 a^{4} b^{2} e^{6} x^{2}-813960 a^{3} b^{3} d \,e^{5} x^{2}+325584 a^{2} b^{4} d^{2} e^{4} x^{2}-76608 a \,b^{5} d^{3} e^{3} x^{2}+8064 b^{6} d^{4} e^{2} x^{2}+646646 a^{5} b \,e^{6} x -587860 a^{4} b^{2} d \,e^{5} x +361760 a^{3} b^{3} d^{2} e^{4} x -144704 a^{2} b^{4} d^{3} e^{3} x +34048 a \,b^{5} d^{4} e^{2} x -3584 b^{6} d^{5} e x +138567 e^{6} a^{6}-184756 d \,e^{5} a^{5} b +167960 d^{2} e^{4} a^{4} b^{2}-103360 d^{3} e^{3} a^{3} b^{3}+41344 d^{4} e^{2} a^{2} b^{4}-9728 d^{5} e a \,b^{5}+1024 d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{969969 e^{7} \left (b x +a \right )^{5}}\)	\(393\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (51051 e^{9} b^{6} x^{9}+342342 a \,b^{5} e^{9} x^{8}+117117 b^{6} d \,e^{8} x^{8}+969969 a^{2} b^{4} e^{9} x^{7}+798798 a \,b^{5} d \,e^{8} x^{7}+69069 b^{6} d^{2} e^{7} x^{7}+1492260 a^{3} b^{3} e^{9} x^{6}+2313003 a^{2} b^{4} d \,e^{8} x^{6}+482790 a \,b^{5} d^{2} e^{7} x^{6}+231 b^{6} d^{3} e^{6} x^{6}+1322685 a^{4} b^{2} e^{9} x^{5}+3662820 a^{3} b^{3} d \,e^{8} x^{5}+1444779 a^{2} b^{4} d^{2} e^{7} x^{5}+2394 a \,b^{5} d^{3} e^{6} x^{5}-252 b^{6} d^{4} e^{5} x^{5}+646646 a^{5} b \,e^{9} x^{4}+3380195 a^{4} b^{2} d \,e^{8} x^{4}+2396660 a^{3} b^{3} d^{2} e^{7} x^{4}+11305 a^{2} b^{4} d^{3} e^{6} x^{4}-2660 a \,b^{5} d^{4} e^{5} x^{4}+280 b^{6} d^{5} e^{4} x^{4}+138567 a^{6} e^{9} x^{3}+1755182 a^{5} b d \,e^{8} x^{3}+2372435 a^{4} b^{2} d^{2} e^{7} x^{3}+32300 a^{3} b^{3} d^{3} e^{6} x^{3}-12920 a^{2} b^{4} d^{4} e^{5} x^{3}+3040 a \,b^{5} d^{5} e^{4} x^{3}-320 b^{6} d^{6} e^{3} x^{3}+415701 a^{6} d \,e^{8} x^{2}+1385670 a^{5} b \,d^{2} e^{7} x^{2}+62985 a^{4} b^{2} d^{3} e^{6} x^{2}-38760 a^{3} b^{3} d^{4} e^{5} x^{2}+15504 a^{2} b^{4} d^{5} e^{4} x^{2}-3648 a \,b^{5} d^{6} e^{3} x^{2}+384 b^{6} d^{7} e^{2} x^{2}+415701 a^{6} d^{2} e^{7} x +92378 a^{5} b \,d^{3} e^{6} x -83980 a^{4} b^{2} d^{4} e^{5} x +51680 a^{3} b^{3} d^{5} e^{4} x -20672 a^{2} b^{4} d^{6} e^{3} x +4864 a \,b^{5} d^{7} e^{2} x -512 b^{6} d^{8} e x +138567 a^{6} d^{3} e^{6}-184756 a^{5} b \,d^{4} e^{5}+167960 a^{4} b^{2} d^{5} e^{4}-103360 a^{3} b^{3} d^{6} e^{3}+41344 a^{2} b^{4} d^{7} e^{2}-9728 a \,b^{5} d^{8} e +1024 b^{6} d^{9}\right ) \sqrt {e x +d}}{969969 \left (b x +a \right ) e^{7}}\)	\(716\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/969969*(e*x+d)^(7/2)*(51051*b^6*e^6*x^6+342342*a*b^5*e^6*x^5-36036*b^6*d*e^5*x^5+969969*a^2*b^4*e^6*x^4-2282
28*a*b^5*d*e^5*x^4+24024*b^6*d^2*e^4*x^4+1492260*a^3*b^3*e^6*x^3-596904*a^2*b^4*d*e^5*x^3+140448*a*b^5*d^2*e^4
*x^3-14784*b^6*d^3*e^3*x^3+1322685*a^4*b^2*e^6*x^2-813960*a^3*b^3*d*e^5*x^2+325584*a^2*b^4*d^2*e^4*x^2-76608*a
*b^5*d^3*e^3*x^2+8064*b^6*d^4*e^2*x^2+646646*a^5*b*e^6*x-587860*a^4*b^2*d*e^5*x+361760*a^3*b^3*d^2*e^4*x-14470
4*a^2*b^4*d^3*e^3*x+34048*a*b^5*d^4*e^2*x-3584*b^6*d^5*e*x+138567*a^6*e^6-184756*a^5*b*d*e^5+167960*a^4*b^2*d^
2*e^4-103360*a^3*b^3*d^3*e^3+41344*a^2*b^4*d^4*e^2-9728*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a
)^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (277) = 554\).
time = 0.31, size = 1006, normalized size = 2.69 \begin {gather*} \frac {2}{153153} \, {\left (9009 \, b^{5} x^{8} e^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {x e + d} a e^{\left (-6\right )} + \frac {2}{2909907} \, {\left (153153 \, b^{5} x^{9} e^{9} + 3072 \, b^{5} d^{9} - 24320 \, a b^{4} d^{8} e + 82688 \, a^{2} b^{3} d^{7} e^{2} - 155040 \, a^{3} b^{2} d^{6} e^{3} + 167960 \, a^{4} b d^{5} e^{4} - 92378 \, a^{5} d^{4} e^{5} + 9009 \, {\left (39 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 3003 \, {\left (69 \, b^{5} d^{2} e^{7} + 665 \, a b^{4} d e^{8} + 646 \, a^{2} b^{3} e^{9}\right )} x^{7} + 231 \, {\left (3 \, b^{5} d^{3} e^{6} + 5225 \, a b^{4} d^{2} e^{7} + 20026 \, a^{2} b^{3} d e^{8} + 9690 \, a^{3} b^{2} e^{9}\right )} x^{6} - 63 \, {\left (12 \, b^{5} d^{4} e^{5} - 95 \, a b^{4} d^{3} e^{6} - 45866 \, a^{2} b^{3} d^{2} e^{7} - 87210 \, a^{3} b^{2} d e^{8} - 20995 \, a^{4} b e^{9}\right )} x^{5} + 7 \, {\left (120 \, b^{5} d^{5} e^{4} - 950 \, a b^{4} d^{4} e^{5} + 3230 \, a^{2} b^{3} d^{3} e^{6} + 513570 \, a^{3} b^{2} d^{2} e^{7} + 482885 \, a^{4} b d e^{8} + 46189 \, a^{5} e^{9}\right )} x^{4} - {\left (960 \, b^{5} d^{6} e^{3} - 7600 \, a b^{4} d^{5} e^{4} + 25840 \, a^{2} b^{3} d^{4} e^{5} - 48450 \, a^{3} b^{2} d^{3} e^{6} - 2372435 \, a^{4} b d^{2} e^{7} - 877591 \, a^{5} d e^{8}\right )} x^{3} + 3 \, {\left (384 \, b^{5} d^{7} e^{2} - 3040 \, a b^{4} d^{6} e^{3} + 10336 \, a^{2} b^{3} d^{5} e^{4} - 19380 \, a^{3} b^{2} d^{4} e^{5} + 20995 \, a^{4} b d^{3} e^{6} + 230945 \, a^{5} d^{2} e^{7}\right )} x^{2} - {\left (1536 \, b^{5} d^{8} e - 12160 \, a b^{4} d^{7} e^{2} + 41344 \, a^{2} b^{3} d^{6} e^{3} - 77520 \, a^{3} b^{2} d^{5} e^{4} + 83980 \, a^{4} b d^{4} e^{5} - 46189 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt {x e + d} b e^{\left (-7\right )} \end {gather*}

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)^(5/2),x, algorithm="maxima")

[Out]

2/153153*(9009*b^5*x^8*e^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 2
4310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b
^4*d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8
)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^
4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 +
21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60
775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a
^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(x*e + d)*a*e^(-6) + 2/2909907*(153153*b^5*x^
9*e^9 + 3072*b^5*d^9 - 24320*a*b^4*d^8*e + 82688*a^2*b^3*d^7*e^2 - 155040*a^3*b^2*d^6*e^3 + 167960*a^4*b*d^5*e
^4 - 92378*a^5*d^4*e^5 + 9009*(39*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 3003*(69*b^5*d^2*e^7 + 665*a*b^4*d*e^8 + 646
*a^2*b^3*e^9)*x^7 + 231*(3*b^5*d^3*e^6 + 5225*a*b^4*d^2*e^7 + 20026*a^2*b^3*d*e^8 + 9690*a^3*b^2*e^9)*x^6 - 63
*(12*b^5*d^4*e^5 - 95*a*b^4*d^3*e^6 - 45866*a^2*b^3*d^2*e^7 - 87210*a^3*b^2*d*e^8 - 20995*a^4*b*e^9)*x^5 + 7*(
120*b^5*d^5*e^4 - 950*a*b^4*d^4*e^5 + 3230*a^2*b^3*d^3*e^6 + 513570*a^3*b^2*d^2*e^7 + 482885*a^4*b*d*e^8 + 461
89*a^5*e^9)*x^4 - (960*b^5*d^6*e^3 - 7600*a*b^4*d^5*e^4 + 25840*a^2*b^3*d^4*e^5 - 48450*a^3*b^2*d^3*e^6 - 2372
435*a^4*b*d^2*e^7 - 877591*a^5*d*e^8)*x^3 + 3*(384*b^5*d^7*e^2 - 3040*a*b^4*d^6*e^3 + 10336*a^2*b^3*d^5*e^4 -
19380*a^3*b^2*d^4*e^5 + 20995*a^4*b*d^3*e^6 + 230945*a^5*d^2*e^7)*x^2 - (1536*b^5*d^8*e - 12160*a*b^4*d^7*e^2
+ 41344*a^2*b^3*d^6*e^3 - 77520*a^3*b^2*d^5*e^4 + 83980*a^4*b*d^4*e^5 - 46189*a^5*d^3*e^6)*x)*sqrt(x*e + d)*b*
e^(-7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (277) = 554\).
time = 0.53, size = 599, normalized size = 1.60 \begin {gather*} \frac {2}{969969} \, {\left (1024 \, b^{6} d^{9} + {\left (51051 \, b^{6} x^{9} + 342342 \, a b^{5} x^{8} + 969969 \, a^{2} b^{4} x^{7} + 1492260 \, a^{3} b^{3} x^{6} + 1322685 \, a^{4} b^{2} x^{5} + 646646 \, a^{5} b x^{4} + 138567 \, a^{6} x^{3}\right )} e^{9} + {\left (117117 \, b^{6} d x^{8} + 798798 \, a b^{5} d x^{7} + 2313003 \, a^{2} b^{4} d x^{6} + 3662820 \, a^{3} b^{3} d x^{5} + 3380195 \, a^{4} b^{2} d x^{4} + 1755182 \, a^{5} b d x^{3} + 415701 \, a^{6} d x^{2}\right )} e^{8} + {\left (69069 \, b^{6} d^{2} x^{7} + 482790 \, a b^{5} d^{2} x^{6} + 1444779 \, a^{2} b^{4} d^{2} x^{5} + 2396660 \, a^{3} b^{3} d^{2} x^{4} + 2372435 \, a^{4} b^{2} d^{2} x^{3} + 1385670 \, a^{5} b d^{2} x^{2} + 415701 \, a^{6} d^{2} x\right )} e^{7} + {\left (231 \, b^{6} d^{3} x^{6} + 2394 \, a b^{5} d^{3} x^{5} + 11305 \, a^{2} b^{4} d^{3} x^{4} + 32300 \, a^{3} b^{3} d^{3} x^{3} + 62985 \, a^{4} b^{2} d^{3} x^{2} + 92378 \, a^{5} b d^{3} x + 138567 \, a^{6} d^{3}\right )} e^{6} - 4 \, {\left (63 \, b^{6} d^{4} x^{5} + 665 \, a b^{5} d^{4} x^{4} + 3230 \, a^{2} b^{4} d^{4} x^{3} + 9690 \, a^{3} b^{3} d^{4} x^{2} + 20995 \, a^{4} b^{2} d^{4} x + 46189 \, a^{5} b d^{4}\right )} e^{5} + 8 \, {\left (35 \, b^{6} d^{5} x^{4} + 380 \, a b^{5} d^{5} x^{3} + 1938 \, a^{2} b^{4} d^{5} x^{2} + 6460 \, a^{3} b^{3} d^{5} x + 20995 \, a^{4} b^{2} d^{5}\right )} e^{4} - 64 \, {\left (5 \, b^{6} d^{6} x^{3} + 57 \, a b^{5} d^{6} x^{2} + 323 \, a^{2} b^{4} d^{6} x + 1615 \, a^{3} b^{3} d^{6}\right )} e^{3} + 128 \, {\left (3 \, b^{6} d^{7} x^{2} + 38 \, a b^{5} d^{7} x + 323 \, a^{2} b^{4} d^{7}\right )} e^{2} - 512 \, {\left (b^{6} d^{8} x + 19 \, a b^{5} d^{8}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}

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)^(5/2),x, algorithm="fricas")

[Out]

2/969969*(1024*b^6*d^9 + (51051*b^6*x^9 + 342342*a*b^5*x^8 + 969969*a^2*b^4*x^7 + 1492260*a^3*b^3*x^6 + 132268
5*a^4*b^2*x^5 + 646646*a^5*b*x^4 + 138567*a^6*x^3)*e^9 + (117117*b^6*d*x^8 + 798798*a*b^5*d*x^7 + 2313003*a^2*
b^4*d*x^6 + 3662820*a^3*b^3*d*x^5 + 3380195*a^4*b^2*d*x^4 + 1755182*a^5*b*d*x^3 + 415701*a^6*d*x^2)*e^8 + (690
69*b^6*d^2*x^7 + 482790*a*b^5*d^2*x^6 + 1444779*a^2*b^4*d^2*x^5 + 2396660*a^3*b^3*d^2*x^4 + 2372435*a^4*b^2*d^
2*x^3 + 1385670*a^5*b*d^2*x^2 + 415701*a^6*d^2*x)*e^7 + (231*b^6*d^3*x^6 + 2394*a*b^5*d^3*x^5 + 11305*a^2*b^4*
d^3*x^4 + 32300*a^3*b^3*d^3*x^3 + 62985*a^4*b^2*d^3*x^2 + 92378*a^5*b*d^3*x + 138567*a^6*d^3)*e^6 - 4*(63*b^6*
d^4*x^5 + 665*a*b^5*d^4*x^4 + 3230*a^2*b^4*d^4*x^3 + 9690*a^3*b^3*d^4*x^2 + 20995*a^4*b^2*d^4*x + 46189*a^5*b*
d^4)*e^5 + 8*(35*b^6*d^5*x^4 + 380*a*b^5*d^5*x^3 + 1938*a^2*b^4*d^5*x^2 + 6460*a^3*b^3*d^5*x + 20995*a^4*b^2*d
^5)*e^4 - 64*(5*b^6*d^6*x^3 + 57*a*b^5*d^6*x^2 + 323*a^2*b^4*d^6*x + 1615*a^3*b^3*d^6)*e^3 + 128*(3*b^6*d^7*x^
2 + 38*a*b^5*d^7*x + 323*a^2*b^4*d^7)*e^2 - 512*(b^6*d^8*x + 19*a*b^5*d^8)*e)*sqrt(x*e + d)*e^(-7)

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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 )^{\frac {5}{2}} \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)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2339 vs. \(2 (277) = 554\).
time = 2.28, size = 2339, normalized size = 6.25 \begin {gather*} \text {Too large to display} \end {gather*}

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)^(5/2),x, algorithm="giac")

[Out]

2/4849845*(9699690*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d^3*e^(-1)*sgn(b*x + a) + 4849845*(3*(x*e + d)^
(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*d^3*e^(-2)*sgn(b*x + a) + 2771340*(5*(x*e + d)^(7
/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*b^3*d^3*e^(-3)*sgn(b*x + a) +
230945*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*s
qrt(x*e + d)*d^4)*a^2*b^4*d^3*e^(-4)*sgn(b*x + a) + 41990*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(
x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*b^5*d^3*e^
(-5)*sgn(b*x + a) + 1615*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*
e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^6*d^3*e^(-6
)*sgn(b*x + a) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b*d^2*e^(-1)*sg
n(b*x + a) + 6235515*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3
)*a^4*b^2*d^2*e^(-2)*sgn(b*x + a) + 923780*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d
^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^3*b^3*d^2*e^(-3)*sgn(b*x + a) + 314925*(63*(x*e + d)^(
11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4
- 693*sqrt(x*e + d)*d^5)*a^2*b^4*d^2*e^(-4)*sgn(b*x + a) + 29070*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)
*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5
+ 3003*sqrt(x*e + d)*d^6)*a*b^5*d^2*e^(-5)*sgn(b*x + a) + 2261*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*
d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)
*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^6*d^2*e^(-6)*sgn(b*x + a) + 4849845*sqrt(x*e + d)
*a^6*d^3*sgn(b*x + a) + 4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*d^2*sgn(b*x + a) + 2494206*(5*(x*e +
d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^5*b*d*e^(-1)*sgn(b*x + a)
+ 692835*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315
*sqrt(x*e + d)*d^4)*a^4*b^2*d*e^(-2)*sgn(b*x + a) + 419900*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*
(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^3*b^3*d*e
^(-3)*sgn(b*x + a) + 72675*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(
x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a^2*b^4*d*e
^(-4)*sgn(b*x + a) + 13566*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 2502
5*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 64
35*sqrt(x*e + d)*d^7)*a*b^5*d*e^(-5)*sgn(b*x + a) + 133*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 23
5620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*
d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b^6*d*e^(-6)*sgn(b*x
+ a) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6*d*sgn(b*x + a) + 92378*(3
5*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e +
d)*d^4)*a^5*b*e^(-1)*sgn(b*x + a) + 104975*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)
*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^4*b^2*e^(-2)*sgn(b*x + a
) + 32300*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^
3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a^3*b^3*e^(-3)*sgn(b*x + a)
+ 11305*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d
^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^
7)*a^2*b^4*e^(-4)*sgn(b*x + a) + 266*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/
2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e +
d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*a*b^5*e^(-5)*sgn(b*x + a) + 21*(12155*(
x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 264
5370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2
)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*b^6*e^(-6)*sgn(b*x + a) + 138567*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^6*sgn(b*x + a))*e^(-1)

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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 )}^{5/2}\,{\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)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

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

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