3.2043 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=100 \[ -\frac{6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac{2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac{2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac{2 b^3 (d+e x)^{13/2}}{13 e^4} \]

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^4) + (2*b*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^4) - (6*b^2*(b*d - a*e)*
(d + e*x)^(11/2))/(11*e^4) + (2*b^3*(d + e*x)^(13/2))/(13*e^4)

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Rubi [A]  time = 0.0365831, 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} \[ -\frac{6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac{2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac{2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac{2 b^3 (d+e x)^{13/2}}{13 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^4) + (2*b*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^4) - (6*b^2*(b*d - a*e)*
(d + e*x)^(11/2))/(11*e^4) + (2*b^3*(d + e*x)^(13/2))/(13*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)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^3 (d+e x)^{5/2}}{e^3}+\frac{3 b (b d-a e)^2 (d+e x)^{7/2}}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^{9/2}}{e^3}+\frac{b^3 (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (d+e x)^{7/2}}{7 e^4}+\frac{2 b (b d-a e)^2 (d+e x)^{9/2}}{3 e^4}-\frac{6 b^2 (b d-a e) (d+e x)^{11/2}}{11 e^4}+\frac{2 b^3 (d+e x)^{13/2}}{13 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0667094, size = 79, normalized size = 0.79 \[ \frac{2 (d+e x)^{7/2} \left (-819 b^2 (d+e x)^2 (b d-a e)+1001 b (d+e x) (b d-a e)^2-429 (b d-a e)^3+231 b^3 (d+e x)^3\right )}{3003 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(-429*(b*d - a*e)^3 + 1001*b*(b*d - a*e)^2*(d + e*x) - 819*b^2*(b*d - a*e)*(d + e*x)^2 + 23
1*b^3*(d + e*x)^3))/(3003*e^4)

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Maple [A]  time = 0.007, size = 116, normalized size = 1.2 \begin{align*}{\frac{462\,{x}^{3}{b}^{3}{e}^{3}+1638\,{x}^{2}a{b}^{2}{e}^{3}-252\,{x}^{2}{b}^{3}d{e}^{2}+2002\,x{a}^{2}b{e}^{3}-728\,xa{b}^{2}d{e}^{2}+112\,x{b}^{3}{d}^{2}e+858\,{e}^{3}{a}^{3}-572\,d{e}^{2}{a}^{2}b+208\,a{d}^{2}e{b}^{2}-32\,{d}^{3}{b}^{3}}{3003\,{e}^{4}} \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),x)

[Out]

2/3003*(e*x+d)^(7/2)*(231*b^3*e^3*x^3+819*a*b^2*e^3*x^2-126*b^3*d*e^2*x^2+1001*a^2*b*e^3*x-364*a*b^2*d*e^2*x+5
6*b^3*d^2*e*x+429*a^3*e^3-286*a^2*b*d*e^2+104*a*b^2*d^2*e-16*b^3*d^3)/e^4

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Maxima [A]  time = 0.961401, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (231 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{3} - 819 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1001 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 429 \,{\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{7}{2}}\right )}}{3003 \, e^{4}} \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),x, algorithm="maxima")

[Out]

2/3003*(231*(e*x + d)^(13/2)*b^3 - 819*(b^3*d - a*b^2*e)*(e*x + d)^(11/2) + 1001*(b^3*d^2 - 2*a*b^2*d*e + a^2*
b*e^2)*(e*x + d)^(9/2) - 429*(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)^(7/2))/e^4

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Fricas [B]  time = 1.17496, size = 595, normalized size = 5.95 \begin{align*} \frac{2 \,{\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \,{\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \,{\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} +{\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{3003 \, e^{4}} \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),x, algorithm="fricas")

[Out]

2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429*a^3*d^3*e^3 + 63*(9*b^3*d*e^5
+ 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e^4 + 299*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*
d^2*e^4 + 2717*a^2*b*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2*d^3*e^3 - 715*a^2*b*d^2*e^4 - 429*
a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2*d^4*e^2 + 143*a^2*b*d^3*e^3 + 1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 4.56432, size = 549, normalized size = 5.49 \begin{align*} \begin{cases} \frac{2 a^{3} d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a^{3} d^{2} x \sqrt{d + e x}}{7} + \frac{6 a^{3} d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a^{3} e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 a^{2} b d^{4} \sqrt{d + e x}}{21 e^{2}} + \frac{2 a^{2} b d^{3} x \sqrt{d + e x}}{21 e} + \frac{10 a^{2} b d^{2} x^{2} \sqrt{d + e x}}{7} + \frac{38 a^{2} b d e x^{3} \sqrt{d + e x}}{21} + \frac{2 a^{2} b e^{2} x^{4} \sqrt{d + e x}}{3} + \frac{16 a b^{2} d^{5} \sqrt{d + e x}}{231 e^{3}} - \frac{8 a b^{2} d^{4} x \sqrt{d + e x}}{231 e^{2}} + \frac{2 a b^{2} d^{3} x^{2} \sqrt{d + e x}}{77 e} + \frac{226 a b^{2} d^{2} x^{3} \sqrt{d + e x}}{231} + \frac{46 a b^{2} d e x^{4} \sqrt{d + e x}}{33} + \frac{6 a b^{2} e^{2} x^{5} \sqrt{d + e x}}{11} - \frac{32 b^{3} d^{6} \sqrt{d + e x}}{3003 e^{4}} + \frac{16 b^{3} d^{5} x \sqrt{d + e x}}{3003 e^{3}} - \frac{4 b^{3} d^{4} x^{2} \sqrt{d + e x}}{1001 e^{2}} + \frac{10 b^{3} d^{3} x^{3} \sqrt{d + e x}}{3003 e} + \frac{106 b^{3} d^{2} x^{4} \sqrt{d + e x}}{429} + \frac{54 b^{3} d e x^{5} \sqrt{d + e x}}{143} + \frac{2 b^{3} e^{2} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{5}{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)**(5/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*a**3*d**3*sqrt(d + e*x)/(7*e) + 6*a**3*d**2*x*sqrt(d + e*x)/7 + 6*a**3*d*e*x**2*sqrt(d + e*x)/7 +
2*a**3*e**2*x**3*sqrt(d + e*x)/7 - 4*a**2*b*d**4*sqrt(d + e*x)/(21*e**2) + 2*a**2*b*d**3*x*sqrt(d + e*x)/(21*
e) + 10*a**2*b*d**2*x**2*sqrt(d + e*x)/7 + 38*a**2*b*d*e*x**3*sqrt(d + e*x)/21 + 2*a**2*b*e**2*x**4*sqrt(d + e
*x)/3 + 16*a*b**2*d**5*sqrt(d + e*x)/(231*e**3) - 8*a*b**2*d**4*x*sqrt(d + e*x)/(231*e**2) + 2*a*b**2*d**3*x**
2*sqrt(d + e*x)/(77*e) + 226*a*b**2*d**2*x**3*sqrt(d + e*x)/231 + 46*a*b**2*d*e*x**4*sqrt(d + e*x)/33 + 6*a*b*
*2*e**2*x**5*sqrt(d + e*x)/11 - 32*b**3*d**6*sqrt(d + e*x)/(3003*e**4) + 16*b**3*d**5*x*sqrt(d + e*x)/(3003*e*
*3) - 4*b**3*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 10*b**3*d**3*x**3*sqrt(d + e*x)/(3003*e) + 106*b**3*d**2*x*
*4*sqrt(d + e*x)/429 + 54*b**3*d*e*x**5*sqrt(d + e*x)/143 + 2*b**3*e**2*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**
(5/2)*(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 = 1.1466, size = 813, normalized size = 8.13 \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),x, algorithm="giac")

[Out]

2/45045*(9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*d^2*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^2*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^2*e^(-3) + 15015*(x*e + d)^(3/2)*a^3*d^2 + 2574*(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*e^(-1) + 858*(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*b^2*d*e^(-2) + 26*(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*e^(-3) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*d + 429*(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*e^(-1) + 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)*a*b^2*e^(-2)
+ 5*(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*e^(-3) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)
*d + 35*(x*e + d)^(3/2)*d^2)*a^3)*e^(-1)