3.1835 \(\int (A+B x) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=164 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)} \]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^3
*(a + b*x)) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(11*e^3*(a + b*x)) + (2*b*B*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(13*e^3*(a + b*x))

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Rubi [A]  time = 0.283811, 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 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^(7/2)*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^3
*(a + b*x)) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(11*e^3*(a + b*x)) + (2*b*B*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(13*e^3*(a + b*x))

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Rubi in Sympy [A]  time = 27.5898, size = 170, normalized size = 1.04 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{9}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{13 b e} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (13 A b e - 9 B a e - 4 B b d\right )}{143 b e^{2}} + \frac{4 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (13 A b e - 9 B a e - 4 B b d\right )}{1287 b e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)*((b*x+a)**2)**(1/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(9/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(13*b*e) + 2*(
d + e*x)**(9/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(13*A*b*e - 9*B*a*e - 4*B*b*d)/
(143*b*e**2) + 4*(d + e*x)**(9/2)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(
13*A*b*e - 9*B*a*e - 4*B*b*d)/(1287*b*e**3*(a + b*x))

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Mathematica [A]  time = 0.181533, size = 88, normalized size = 0.54 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{9/2} \left (13 a e (11 A e-2 B d+9 B e x)+13 A b e (9 e x-2 d)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(9/2)*(13*A*b*e*(-2*d + 9*e*x) + 13*a*e*(-2*B*d +
11*A*e + 9*B*e*x) + b*B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3*(a + b*x))

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Maple [A]  time = 0.009, size = 89, normalized size = 0.5 \[{\frac{198\,B{x}^{2}b{e}^{2}+234\,Ab{e}^{2}x+234\,aB{e}^{2}x-72\,Bbdex+286\,A{e}^{2}a-52\,Abde-52\,aBde+16\,Bb{d}^{2}}{1287\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2),x)

[Out]

2/1287*(e*x+d)^(9/2)*(99*B*b*e^2*x^2+117*A*b*e^2*x+117*B*a*e^2*x-36*B*b*d*e*x+14
3*A*a*e^2-26*A*b*d*e-26*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 = 0.734361, size = 355, normalized size = 2.16 \[ \frac{2 \,{\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e +{\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \,{\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \,{\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} +{\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt{e x + d} A}{99 \, e^{2}} + \frac{2 \,{\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \,{\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \,{\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \,{\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \,{\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} -{\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt{e x + d} B}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/99*(9*b*e^5*x^5 - 2*b*d^5 + 11*a*d^4*e + (34*b*d*e^4 + 11*a*e^5)*x^4 + 2*(23*b
*d^2*e^3 + 22*a*d*e^4)*x^3 + 6*(4*b*d^3*e^2 + 11*a*d^2*e^3)*x^2 + (b*d^4*e + 44*
a*d^3*e^2)*x)*sqrt(e*x + d)*A/e^2 + 2/1287*(99*b*e^6*x^6 + 8*b*d^6 - 26*a*d^5*e
+ 9*(40*b*d*e^5 + 13*a*e^6)*x^5 + 2*(229*b*d^2*e^4 + 221*a*d*e^5)*x^4 + 2*(106*b
*d^3*e^3 + 299*a*d^2*e^4)*x^3 + 3*(b*d^4*e^2 + 104*a*d^3*e^3)*x^2 - (4*b*d^5*e -
13*a*d^4*e^2)*x)*sqrt(e*x + d)*B/e^3

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Fricas [A]  time = 0.280811, size = 311, normalized size = 1.9 \[ \frac{2 \,{\left (99 \, B b e^{6} x^{6} + 8 \, B b d^{6} + 143 \, A a d^{4} e^{2} - 26 \,{\left (B a + A b\right )} d^{5} e + 9 \,{\left (40 \, B b d e^{5} + 13 \,{\left (B a + A b\right )} e^{6}\right )} x^{5} +{\left (458 \, B b d^{2} e^{4} + 143 \, A a e^{6} + 442 \,{\left (B a + A b\right )} d e^{5}\right )} x^{4} + 2 \,{\left (106 \, B b d^{3} e^{3} + 286 \, A a d e^{5} + 299 \,{\left (B a + A b\right )} d^{2} e^{4}\right )} x^{3} + 3 \,{\left (B b d^{4} e^{2} + 286 \, A a d^{2} e^{4} + 104 \,{\left (B a + A b\right )} d^{3} e^{3}\right )} x^{2} -{\left (4 \, B b d^{5} e - 572 \, A a d^{3} e^{3} - 13 \,{\left (B a + A b\right )} d^{4} e^{2}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/1287*(99*B*b*e^6*x^6 + 8*B*b*d^6 + 143*A*a*d^4*e^2 - 26*(B*a + A*b)*d^5*e + 9*
(40*B*b*d*e^5 + 13*(B*a + A*b)*e^6)*x^5 + (458*B*b*d^2*e^4 + 143*A*a*e^6 + 442*(
B*a + A*b)*d*e^5)*x^4 + 2*(106*B*b*d^3*e^3 + 286*A*a*d*e^5 + 299*(B*a + A*b)*d^2
*e^4)*x^3 + 3*(B*b*d^4*e^2 + 286*A*a*d^2*e^4 + 104*(B*a + A*b)*d^3*e^3)*x^2 - (4
*B*b*d^5*e - 572*A*a*d^3*e^3 - 13*(B*a + A*b)*d^4*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)**(7/2)*((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.309543, 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)^(7/2),x, algorithm="giac")

[Out]

Done