3.1836 \(\int (A+B x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=164 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)} \]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3
*(a + b*x)) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(9*e^3*(a + b*x)) + (2*b*B*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(11*e^3*(a + b*x))

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Rubi [A]  time = 0.260046, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3
*(a + b*x)) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(9*e^3*(a + b*x)) + (2*b*B*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(11*e^3*(a + b*x))

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Rubi in Sympy [A]  time = 27.4679, size = 170, normalized size = 1.04 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{11 b e} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e - 7 B a e - 4 B b d\right )}{99 b e^{2}} + \frac{4 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e - 7 B a e - 4 B b d\right )}{693 b e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(5/2)*((b*x+a)**2)**(1/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(7/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(11*b*e) + 2*(
d + e*x)**(7/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(11*A*b*e - 7*B*a*e - 4*B*b*d)/
(99*b*e**2) + 4*(d + e*x)**(7/2)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(1
1*A*b*e - 7*B*a*e - 4*B*b*d)/(693*b*e**3*(a + b*x))

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Mathematica [A]  time = 0.151243, size = 88, normalized size = 0.54 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(11*A*b*e*(-2*d + 7*e*x) + 11*a*e*(-2*B*d +
9*A*e + 7*B*e*x) + b*B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3*(a + b*x))

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Maple [A]  time = 0.007, size = 89, normalized size = 0.5 \[{\frac{126\,B{x}^{2}b{e}^{2}+154\,Ab{e}^{2}x+154\,aB{e}^{2}x-56\,Bbdex+198\,A{e}^{2}a-44\,Abde-44\,aBde+16\,Bb{d}^{2}}{693\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2),x)

[Out]

2/693*(e*x+d)^(7/2)*(63*B*b*e^2*x^2+77*A*b*e^2*x+77*B*a*e^2*x-28*B*b*d*e*x+99*A*
a*e^2-22*A*b*d*e-22*B*a*d*e+8*B*b*d^2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

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Maxima [A]  time = 0.734237, size = 289, normalized size = 1.76 \[ \frac{2 \,{\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e +{\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \,{\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} +{\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x + d} A}{63 \, e^{2}} + \frac{2 \,{\left (63 \, b e^{5} x^{5} + 8 \, b d^{5} - 22 \, a d^{4} e + 7 \,{\left (23 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} +{\left (113 \, b d^{2} e^{3} + 209 \, a d e^{4}\right )} x^{3} + 3 \,{\left (b d^{3} e^{2} + 55 \, a d^{2} e^{3}\right )} x^{2} -{\left (4 \, b d^{4} e - 11 \, a d^{3} e^{2}\right )} x\right )} \sqrt{e x + d} B}{693 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/63*(7*b*e^4*x^4 - 2*b*d^4 + 9*a*d^3*e + (19*b*d*e^3 + 9*a*e^4)*x^3 + 3*(5*b*d^
2*e^2 + 9*a*d*e^3)*x^2 + (b*d^3*e + 27*a*d^2*e^2)*x)*sqrt(e*x + d)*A/e^2 + 2/693
*(63*b*e^5*x^5 + 8*b*d^5 - 22*a*d^4*e + 7*(23*b*d*e^4 + 11*a*e^5)*x^4 + (113*b*d
^2*e^3 + 209*a*d*e^4)*x^3 + 3*(b*d^3*e^2 + 55*a*d^2*e^3)*x^2 - (4*b*d^4*e - 11*a
*d^3*e^2)*x)*sqrt(e*x + d)*B/e^3

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Fricas [A]  time = 0.278622, size = 255, normalized size = 1.55 \[ \frac{2 \,{\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \,{\left (B a + A b\right )} d^{4} e + 7 \,{\left (23 \, B b d e^{4} + 11 \,{\left (B a + A b\right )} e^{5}\right )} x^{4} +{\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \,{\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \,{\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \,{\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} -{\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \,{\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b*e^5*x^5 + 8*B*b*d^5 + 99*A*a*d^3*e^2 - 22*(B*a + A*b)*d^4*e + 7*(2
3*B*b*d*e^4 + 11*(B*a + A*b)*e^5)*x^4 + (113*B*b*d^2*e^3 + 99*A*a*e^5 + 209*(B*a
+ A*b)*d*e^4)*x^3 + 3*(B*b*d^3*e^2 + 99*A*a*d*e^4 + 55*(B*a + A*b)*d^2*e^3)*x^2
- (4*B*b*d^4*e - 297*A*a*d^2*e^3 - 11*(B*a + A*b)*d^3*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(5/2)*((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.312025, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done