3.1842 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=162 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{5 e^3 (a+b x) (d+e x)^{5/2}}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.256568, antiderivative size = 162, 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 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{5 e^3 (a+b x) (d+e x)^{5/2}}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.664, size = 173, normalized size = 1.07 \[ - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} + \frac{2 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 5 B a e + 4 B b d\right )}{5 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} - \frac{4 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 5 B a e + 4 B b d\right )}{15 e^{3} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a + 2*b*x)*(A*e - B*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e*(d + e*x)**(5/2
)*(a*e - b*d)) + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*e - 5*B*a*e + 4*B*b*d)/
(5*e**2*(d + e*x)**(3/2)*(a*e - b*d)) - 4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*
e - 5*B*a*e + 4*B*b*d)/(15*e**3*(a + b*x)*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.104449, size = 86, normalized size = 0.53 \[ -\frac{2 \sqrt{(a+b x)^2} \left (a e (3 A e+2 B d+5 B e x)+A b e (2 d+5 e x)+b B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(A*b*e*(2*d + 5*e*x) + a*e*(2*B*d + 3*A*e + 5*B*e*x) + b*B
*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(a + b*x)*(d + e*x)^(5/2))

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Maple [A]  time = 0.008, size = 89, normalized size = 0.6 \[ -{\frac{30\,B{x}^{2}b{e}^{2}+10\,Ab{e}^{2}x+10\,aB{e}^{2}x+40\,Bbdex+6\,A{e}^{2}a+4\,Abde+4\,aBde+16\,Bb{d}^{2}}{15\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15/(e*x+d)^(5/2)*(15*B*b*e^2*x^2+5*A*b*e^2*x+5*B*a*e^2*x+20*B*b*d*e*x+3*A*a*e
^2+2*A*b*d*e+2*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 = 0.741041, size = 159, normalized size = 0.98 \[ -\frac{2 \,{\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} A}{15 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )} \sqrt{e x + d}} - \frac{2 \,{\left (15 \, b e^{2} x^{2} + 8 \, b d^{2} + 2 \, a d e + 5 \,{\left (4 \, b d e + a e^{2}\right )} x\right )} B}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

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]

-2/15*(5*b*e*x + 2*b*d + 3*a*e)*A/((e^4*x^2 + 2*d*e^3*x + d^2*e^2)*sqrt(e*x + d)
) - 2/15*(15*b*e^2*x^2 + 8*b*d^2 + 2*a*d*e + 5*(4*b*d*e + a*e^2)*x)*B/((e^5*x^2
+ 2*d*e^4*x + d^2*e^3)*sqrt(e*x + d))

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Fricas [A]  time = 0.287826, size = 122, normalized size = 0.75 \[ -\frac{2 \,{\left (15 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 3 \, A a e^{2} + 2 \,{\left (B a + A b\right )} d e + 5 \,{\left (4 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

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]

-2/15*(15*B*b*e^2*x^2 + 8*B*b*d^2 + 3*A*a*e^2 + 2*(B*a + A*b)*d*e + 5*(4*B*b*d*e
+ (B*a + A*b)*e^2)*x)/((e^5*x^2 + 2*d*e^4*x + d^2*e^3)*sqrt(e*x + d))

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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)*((b*x+a)**2)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29074, size = 182, normalized size = 1.12 \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} B b{\rm sign}\left (b x + a\right ) - 10 \,{\left (x e + d\right )} B b d{\rm sign}\left (b x + a\right ) + 3 \, B b d^{2}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )} B a e{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )} A b e{\rm sign}\left (b x + a\right ) - 3 \, B a d e{\rm sign}\left (b x + a\right ) - 3 \, A b d e{\rm sign}\left (b x + a\right ) + 3 \, A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

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]

-2/15*(15*(x*e + d)^2*B*b*sign(b*x + a) - 10*(x*e + d)*B*b*d*sign(b*x + a) + 3*B
*b*d^2*sign(b*x + a) + 5*(x*e + d)*B*a*e*sign(b*x + a) + 5*(x*e + d)*A*b*e*sign(
b*x + a) - 3*B*a*d*e*sign(b*x + a) - 3*A*b*d*e*sign(b*x + a) + 3*A*a*e^2*sign(b*
x + a))*e^(-3)/(x*e + d)^(5/2)