3.2042 \(\int (a+b x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=100 \[ -\frac{6 b^2 (d+e x)^{13/2} (b d-a e)}{13 e^4}+\frac{6 b (d+e x)^{11/2} (b d-a e)^2}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^3}{9 e^4}+\frac{2 b^3 (d+e x)^{15/2}}{15 e^4} \]

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^4) - (6*b^2*(b*d - a*e
)*(d + e*x)^(13/2))/(13*e^4) + (2*b^3*(d + e*x)^(15/2))/(15*e^4)

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Rubi [A]  time = 0.0440653, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{6 b^2 (d+e x)^{13/2} (b d-a e)}{13 e^4}+\frac{6 b (d+e x)^{11/2} (b d-a e)^2}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^3}{9 e^4}+\frac{2 b^3 (d+e x)^{15/2}}{15 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^4) - (6*b^2*(b*d - a*e
)*(d + e*x)^(13/2))/(13*e^4) + (2*b^3*(d + e*x)^(15/2))/(15*e^4)

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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^3 (d+e x)^{7/2}}{e^3}+\frac{3 b (b d-a e)^2 (d+e x)^{9/2}}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^{11/2}}{e^3}+\frac{b^3 (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (d+e x)^{9/2}}{9 e^4}+\frac{6 b (b d-a e)^2 (d+e x)^{11/2}}{11 e^4}-\frac{6 b^2 (b d-a e) (d+e x)^{13/2}}{13 e^4}+\frac{2 b^3 (d+e x)^{15/2}}{15 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0888466, size = 79, normalized size = 0.79 \[ \frac{2 (d+e x)^{9/2} \left (-1485 b^2 (d+e x)^2 (b d-a e)+1755 b (d+e x) (b d-a e)^2-715 (b d-a e)^3+429 b^3 (d+e x)^3\right )}{6435 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(9/2)*(-715*(b*d - a*e)^3 + 1755*b*(b*d - a*e)^2*(d + e*x) - 1485*b^2*(b*d - a*e)*(d + e*x)^2 + 4
29*b^3*(d + e*x)^3))/(6435*e^4)

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Maple [A]  time = 0.006, size = 116, normalized size = 1.2 \begin{align*}{\frac{858\,{x}^{3}{b}^{3}{e}^{3}+2970\,{x}^{2}a{b}^{2}{e}^{3}-396\,{x}^{2}{b}^{3}d{e}^{2}+3510\,x{a}^{2}b{e}^{3}-1080\,xa{b}^{2}d{e}^{2}+144\,x{b}^{3}{d}^{2}e+1430\,{e}^{3}{a}^{3}-780\,d{e}^{2}{a}^{2}b+240\,a{d}^{2}e{b}^{2}-32\,{d}^{3}{b}^{3}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6435*(e*x+d)^(9/2)*(429*b^3*e^3*x^3+1485*a*b^2*e^3*x^2-198*b^3*d*e^2*x^2+1755*a^2*b*e^3*x-540*a*b^2*d*e^2*x+
72*b^3*d^2*e*x+715*a^3*e^3-390*a^2*b*d*e^2+120*a*b^2*d^2*e-16*b^3*d^3)/e^4

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Maxima [A]  time = 0.981245, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{3} - 1485 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1755 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/6435*(429*(e*x + d)^(15/2)*b^3 - 1485*(b^3*d - a*b^2*e)*(e*x + d)^(13/2) + 1755*(b^3*d^2 - 2*a*b^2*d*e + a^2
*b*e^2)*(e*x + d)^(11/2) - 715*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(9/2))/e^4

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Fricas [B]  time = 1.31748, size = 716, normalized size = 7.16 \begin{align*} \frac{2 \,{\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \,{\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \,{\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \,{\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \,{\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715*a^3*d^4*e^3 + 33*(46*b^3*d*e^
6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2*e^5 + 600*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*
a*b^2*d^2*e^5 + 1326*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a*b^2*d^3*e^4 + 1794*a^2*b*d^2*e^5
+ 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b
^3*d^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 10.9323, size = 654, normalized size = 6.54 \begin{align*} \begin{cases} \frac{2 a^{3} d^{4} \sqrt{d + e x}}{9 e} + \frac{8 a^{3} d^{3} x \sqrt{d + e x}}{9} + \frac{4 a^{3} d^{2} e x^{2} \sqrt{d + e x}}{3} + \frac{8 a^{3} d e^{2} x^{3} \sqrt{d + e x}}{9} + \frac{2 a^{3} e^{3} x^{4} \sqrt{d + e x}}{9} - \frac{4 a^{2} b d^{5} \sqrt{d + e x}}{33 e^{2}} + \frac{2 a^{2} b d^{4} x \sqrt{d + e x}}{33 e} + \frac{16 a^{2} b d^{3} x^{2} \sqrt{d + e x}}{11} + \frac{92 a^{2} b d^{2} e x^{3} \sqrt{d + e x}}{33} + \frac{68 a^{2} b d e^{2} x^{4} \sqrt{d + e x}}{33} + \frac{6 a^{2} b e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 a b^{2} d^{6} \sqrt{d + e x}}{429 e^{3}} - \frac{8 a b^{2} d^{5} x \sqrt{d + e x}}{429 e^{2}} + \frac{2 a b^{2} d^{4} x^{2} \sqrt{d + e x}}{143 e} + \frac{424 a b^{2} d^{3} x^{3} \sqrt{d + e x}}{429} + \frac{916 a b^{2} d^{2} e x^{4} \sqrt{d + e x}}{429} + \frac{240 a b^{2} d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{6 a b^{2} e^{3} x^{6} \sqrt{d + e x}}{13} - \frac{32 b^{3} d^{7} \sqrt{d + e x}}{6435 e^{4}} + \frac{16 b^{3} d^{6} x \sqrt{d + e x}}{6435 e^{3}} - \frac{4 b^{3} d^{5} x^{2} \sqrt{d + e x}}{2145 e^{2}} + \frac{2 b^{3} d^{4} x^{3} \sqrt{d + e x}}{1287 e} + \frac{320 b^{3} d^{3} x^{4} \sqrt{d + e x}}{1287} + \frac{412 b^{3} d^{2} e x^{5} \sqrt{d + e x}}{715} + \frac{92 b^{3} d e^{2} x^{6} \sqrt{d + e x}}{195} + \frac{2 b^{3} e^{3} x^{7} \sqrt{d + e x}}{15} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a**3*d**4*sqrt(d + e*x)/(9*e) + 8*a**3*d**3*x*sqrt(d + e*x)/9 + 4*a**3*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**3*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**3*e**3*x**4*sqrt(d + e*x)/9 - 4*a**2*b*d**5*sqrt(d + e*x)/(33*e*
*2) + 2*a**2*b*d**4*x*sqrt(d + e*x)/(33*e) + 16*a**2*b*d**3*x**2*sqrt(d + e*x)/11 + 92*a**2*b*d**2*e*x**3*sqrt
(d + e*x)/33 + 68*a**2*b*d*e**2*x**4*sqrt(d + e*x)/33 + 6*a**2*b*e**3*x**5*sqrt(d + e*x)/11 + 16*a*b**2*d**6*s
qrt(d + e*x)/(429*e**3) - 8*a*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*a*b**2*d**4*x**2*sqrt(d + e*x)/(143*e)
+ 424*a*b**2*d**3*x**3*sqrt(d + e*x)/429 + 916*a*b**2*d**2*e*x**4*sqrt(d + e*x)/429 + 240*a*b**2*d*e**2*x**5*s
qrt(d + e*x)/143 + 6*a*b**2*e**3*x**6*sqrt(d + e*x)/13 - 32*b**3*d**7*sqrt(d + e*x)/(6435*e**4) + 16*b**3*d**6
*x*sqrt(d + e*x)/(6435*e**3) - 4*b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*b**3*d**4*x**3*sqrt(d + e*x)/(12
87*e) + 320*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 412*b**3*d**2*e*x**5*sqrt(d + e*x)/715 + 92*b**3*d*e**2*x**6*s
qrt(d + e*x)/195 + 2*b**3*e**3*x**7*sqrt(d + e*x)/15, Ne(e, 0)), (d**(7/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*
x**3 + b**3*x**4/4), True))

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Giac [B]  time = 1.18773, size = 1230, normalized size = 12.3 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

2/45045*(9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*d^3*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e
+ d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b^2*d^3*e^(-2) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 18
9*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^3*d^3*e^(-3) + 15015*(x*e + d)^(3/2)*a^3*d^3 + 3861*(15*(x*
e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b*d^2*e^(-1) + 1287*(35*(x*e + d)^(9/2) - 13
5*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b^2*d^2*e^(-2) + 39*(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)*b^3*d^2*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*d^2 + 1287*(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^2*b*d*e^(-1) + 117*(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*b^2*d*e^(-2) + 15*(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)*b^3*d*e^(-3) + 1287*(15*(x*e + d)^(7/2) -
42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*d + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 29
70*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^2*b*e^(-1) + 15*(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*b^2*e^(-2) + (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 + 1501
5*(x*e + d)^(3/2)*d^6)*b^3*e^(-3) + 143*(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^3)*e^(-1)