3.1728 \(\int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(7/2))/(7*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(9*e^4) + (2*b^2*B*(d + e*x)^(11/2))/(11*e^
4)

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Rubi [A]  time = 0.0522237, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(7/2))/(7*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(9*e^4) + (2*b^2*B*(d + e*x)^(11/2))/(11*e^
4)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.102909, size = 107, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (-385 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+495 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-693 (b d-a e)^2 (B d-A e)+315 b^2 B (d+e x)^3\right )}{3465 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(2*(d + e*x)^(5/2)*(-693*(b*d - a*e)^2*(B*d - A*e) + 495*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 3
85*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2 + 315*b^2*B*(d + e*x)^3))/(3465*e^4)

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Maple [A]  time = 0.005, size = 169, normalized size = 1.3 \begin{align*}{\frac{630\,B{b}^{2}{x}^{3}{e}^{3}+770\,A{b}^{2}{e}^{3}{x}^{2}+1540\,Bab{e}^{3}{x}^{2}-420\,B{b}^{2}d{e}^{2}{x}^{2}+1980\,Aab{e}^{3}x-440\,A{b}^{2}d{e}^{2}x+990\,B{a}^{2}{e}^{3}x-880\,Babd{e}^{2}x+240\,B{b}^{2}{d}^{2}ex+1386\,{a}^{2}A{e}^{3}-792\,Aabd{e}^{2}+176\,A{b}^{2}{d}^{2}e-396\,B{a}^{2}d{e}^{2}+352\,Bab{d}^{2}e-96\,B{b}^{2}{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x)

[Out]

2/3465*(e*x+d)^(5/2)*(315*B*b^2*e^3*x^3+385*A*b^2*e^3*x^2+770*B*a*b*e^3*x^2-210*B*b^2*d*e^2*x^2+990*A*a*b*e^3*
x-220*A*b^2*d*e^2*x+495*B*a^2*e^3*x-440*B*a*b*d*e^2*x+120*B*b^2*d^2*e*x+693*A*a^2*e^3-396*A*a*b*d*e^2+88*A*b^2
*d^2*e-198*B*a^2*d*e^2+176*B*a*b*d^2*e-48*B*b^2*d^3)/e^4

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Maxima [A]  time = 1.95026, size = 215, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{2} - 385 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*b^2 - 385*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(9/2) + 495*(3*B*b^2*d^2
- 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x + d)^(7/2) - 693*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A
*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(5/2))/e^4

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Fricas [B]  time = 1.54376, size = 652, normalized size = 5.09 \begin{align*} \frac{2 \,{\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \,{\left (12 \, B b^{2} d e^{4} + 11 \,{\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (3 \, B b^{2} d^{2} e^{3} + 110 \,{\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \,{\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} +{\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^2*e^5*x^5 - 48*B*b^2*d^5 + 693*A*a^2*d^2*e^3 + 88*(2*B*a*b + A*b^2)*d^4*e - 198*(B*a^2 + 2*A*a
*b)*d^3*e^2 + 35*(12*B*b^2*d*e^4 + 11*(2*B*a*b + A*b^2)*e^5)*x^4 + 5*(3*B*b^2*d^2*e^3 + 110*(2*B*a*b + A*b^2)*
d*e^4 + 99*(B*a^2 + 2*A*a*b)*e^5)*x^3 - 3*(6*B*b^2*d^3*e^2 - 231*A*a^2*e^5 - 11*(2*B*a*b + A*b^2)*d^2*e^3 - 26
4*(B*a^2 + 2*A*a*b)*d*e^4)*x^2 + (24*B*b^2*d^4*e + 1386*A*a^2*d*e^4 - 44*(2*B*a*b + A*b^2)*d^3*e^2 + 99*(B*a^2
+ 2*A*a*b)*d^2*e^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 16.598, size = 586, normalized size = 4.58 \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)**2*(B*x+A)*(e*x+d)**(3/2),x)

[Out]

A*a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a**2*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 4*A*a*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*A*a*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*A*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d
+ e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*A*b**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**3 + 2*B*a**2*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e*
*2 + 2*B*a**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*B*a*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**3 + 4*B*a*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**3 + 2*B*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**4 + 2*B*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**4

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Giac [B]  time = 1.60918, size = 698, normalized size = 5.45 \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)^2*(B*x+A)*(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^2*d*e^(-1) + 462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3
/2)*d)*A*a*b*d*e^(-1) + 66*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a*b*d*e^(-2)
+ 33*(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*e^(-2) + 11*(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)*B*b^2*d*e^(-3) + 1155*(x*e +
d)^(3/2)*A*a^2*d + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^2*e^(-1) + 66*
(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*b*e^(-1) + 22*(35*(x*e + d)^(9/2) - 1
35*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a*b*e^(-2) + 11*(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*e^(-2) + (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*b^2*e^(-3) + 231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^2)*e^(-1)