3.1786 \(\int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

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 = 0.160892, antiderivative size = 128, 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 \[ -\frac{2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{2 b^2 B (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 54.7848, size = 126, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{9}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{9 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{5 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.227155, size = 139, normalized size = 1.09 \[ \frac{2 (d+e x)^{5/2} \left (99 a^2 e^2 (7 A e-2 B d+5 B e x)+22 a b e \left (9 A e (5 e x-2 d)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^2 \left (11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(99*a^2*e^2*(-2*B*d + 7*A*e + 5*B*e*x) + 22*a*b*e*(9*A*e*(-2*
d + 5*e*x) + B*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) + b^2*(11*A*e*(8*d^2 - 20*d*e*x
+ 35*e^2*x^2) - 3*B*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/(3465*
e^4)

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Maple [A]  time = 0.012, size = 169, normalized size = 1.3 \[{\frac{630\,B{x}^{3}{b}^{2}{e}^{3}+770\,A{b}^{2}{e}^{3}{x}^{2}+1540\,Bab{e}^{3}{x}^{2}-420\,B{b}^{2}d{e}^{2}{x}^{2}+1980\,Axab{e}^{3}-440\,Ax{b}^{2}d{e}^{2}+990\,Bx{a}^{2}{e}^{3}-880\,Bxabd{e}^{2}+240\,B{b}^{2}{d}^{2}ex+1386\,A{a}^{2}{e}^{3}-792\,Aabd{e}^{2}+176\,A{b}^{2}{d}^{2}e-396\,Bd{e}^{2}{a}^{2}+352\,B{d}^{2}abe-96\,B{b}^{2}{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^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 = 0.733934, size = 215, normalized size = 1.68 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{2} - 385 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*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 [A]  time = 0.314826, size = 390, normalized size = 3.05 \[ \frac{2 \,{\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \,{\left (12 \, B b^{2} d e^{4} + 11 \,{\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (3 \, B b^{2} d^{2} e^{3} + 110 \,{\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \,{\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} +{\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*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 - 264*(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 = 10.7695, size = 586, normalized size = 4.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**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/XCAS [A]  time = 0.317867, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done