3.1713 \(\int \frac{(a+b x)^2 (A+B x)}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*Sqrt[d + e*x])/e^4 + (2*(b*d - a*e)*(3*b*B*d - 2*A
*b*e - a*B*e)*(d + e*x)^(3/2))/(3*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e
*x)^(5/2))/(5*e^4) + (2*b^2*B*(d + e*x)^(7/2))/(7*e^4)

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Rubi [A]  time = 0.152867, antiderivative size = 126, 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 \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*Sqrt[d + e*x])/e^4 + (2*(b*d - a*e)*(3*b*B*d - 2*A
*b*e - a*B*e)*(d + e*x)^(3/2))/(3*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e
*x)^(5/2))/(5*e^4) + (2*b^2*B*(d + e*x)^(7/2))/(7*e^4)

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Rubi in Sympy [A]  time = 29.8856, size = 124, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{5}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.188579, size = 137, normalized size = 1.09 \[ \frac{2 \sqrt{d+e x} \left (35 a^2 e^2 (3 A e-2 B d+B e x)+14 a b e \left (5 A e (e x-2 d)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+b^2 \left (7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{105 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(35*a^2*e^2*(-2*B*d + 3*A*e + B*e*x) + 14*a*b*e*(5*A*e*(-2*d +
e*x) + B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) + b^2*(7*A*e*(8*d^2 - 4*d*e*x + 3*e^2*x^
2) - 3*B*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3))))/(105*e^4)

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Maple [A]  time = 0.01, size = 169, normalized size = 1.3 \[{\frac{30\,B{b}^{2}{x}^{3}{e}^{3}+42\,A{b}^{2}{e}^{3}{x}^{2}+84\,Bab{e}^{3}{x}^{2}-36\,B{b}^{2}d{e}^{2}{x}^{2}+140\,Aab{e}^{3}x-56\,A{b}^{2}d{e}^{2}x+70\,B{a}^{2}{e}^{3}x-112\,Babd{e}^{2}x+48\,B{b}^{2}{d}^{2}ex+210\,{a}^{2}A{e}^{3}-280\,Aabd{e}^{2}+112\,A{b}^{2}{d}^{2}e-140\,B{a}^{2}d{e}^{2}+224\,Bab{d}^{2}e-96\,B{b}^{2}{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(B*x+A)/(e*x+d)^(1/2),x)

[Out]

2/105*(e*x+d)^(1/2)*(15*B*b^2*e^3*x^3+21*A*b^2*e^3*x^2+42*B*a*b*e^3*x^2-18*B*b^2
*d*e^2*x^2+70*A*a*b*e^3*x-28*A*b^2*d*e^2*x+35*B*a^2*e^3*x-56*B*a*b*d*e^2*x+24*B*
b^2*d^2*e*x+105*A*a^2*e^3-140*A*a*b*d*e^2+56*A*b^2*d^2*e-70*B*a^2*d*e^2+112*B*a*
b*d^2*e-48*B*b^2*d^3)/e^4

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Maxima [A]  time = 1.34719, size = 215, normalized size = 1.71 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{2} - 21 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*(e*x + d)^(7/2)*B*b^2 - 21*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)
^(5/2) + 35*(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)^(3/2) - 105*(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)*sqrt(e*x + d))/e^4

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Fricas [A]  time = 0.226124, size = 209, normalized size = 1.66 \[ \frac{2 \,{\left (15 \, B b^{2} e^{3} x^{3} - 48 \, B b^{2} d^{3} + 105 \, A a^{2} e^{3} + 56 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 70 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (6 \, B b^{2} d e^{2} - 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 28 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*b^2*e^3*x^3 - 48*B*b^2*d^3 + 105*A*a^2*e^3 + 56*(2*B*a*b + A*b^2)*d^
2*e - 70*(B*a^2 + 2*A*a*b)*d*e^2 - 3*(6*B*b^2*d*e^2 - 7*(2*B*a*b + A*b^2)*e^3)*x
^2 + (24*B*b^2*d^2*e - 28*(2*B*a*b + A*b^2)*d*e^2 + 35*(B*a^2 + 2*A*a*b)*e^3)*x)
*sqrt(e*x + d)/e^4

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Sympy [A]  time = 34.2601, size = 583, normalized size = 4.63 \[ \begin{cases} - \frac{\frac{2 A a^{2} d}{\sqrt{d + e x}} + 2 A a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 A a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 A a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 A b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a^{2} d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{4 B a b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 B a b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B b^{2} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B b^{2} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{A a^{2} x + \frac{B b^{2} x^{4}}{4} + \frac{x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac{x^{2} \left (2 A a b + B a^{2}\right )}{2}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22447, size = 317, normalized size = 2.52 \[ \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{2} e^{\left (-1\right )} + 70 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a b e^{\left (-1\right )} + 14 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B a b e^{\left (-10\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A b^{2} e^{\left (-10\right )} + 3 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B b^{2} e^{\left (-21\right )} + 105 \, \sqrt{x e + d} A a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*e^(-1) + 70*((x*e + d)^(3/
2) - 3*sqrt(x*e + d)*d)*A*a*b*e^(-1) + 14*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^
(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*B*a*b*e^(-10) + 7*(3*(x*e + d)^(5/2)*e^8
- 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*A*b^2*e^(-10) + 3*(5*(x*
e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^18 - 35
*sqrt(x*e + d)*d^3*e^18)*B*b^2*e^(-21) + 105*sqrt(x*e + d)*A*a^2)*e^(-1)