3.1966 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]

[Out]

-((b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)*(d + e*x)^6) + (
2*b*(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) - (
b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3*(a + b*x)*(d + e*x)^4)

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Rubi [A]  time = 0.220937, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)*(d + e*x)^6) + (
2*b*(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) - (
b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3*(a + b*x)*(d + e*x)^4)

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Rubi in Sympy [A]  time = 28.1736, size = 114, normalized size = 0.78 \[ - \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 e^{2} \left (d + e x\right )^{5}} + \frac{b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{60 e^{3} \left (a + b x\right ) \left (d + e x\right )^{5}} - \frac{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e \left (d + e x\right )^{6}} \]

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,x)

[Out]

-b*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(12*e**2*(d + e*x)**5) + b*(a*e - b*d)*sqrt(
a**2 + 2*a*b*x + b**2*x**2)/(60*e**3*(a + b*x)*(d + e*x)**5) - (a + b*x)*sqrt(a*
*2 + 2*a*b*x + b**2*x**2)/(6*e*(d + e*x)**6)

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Mathematica [A]  time = 0.0571813, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*
e^2*x^2)))/(60*e^3*(a + b*x)*(d + e*x)^6)

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Maple [A]  time = 0.012, size = 78, normalized size = 0.5 \[ -{\frac{15\,{x}^{2}{b}^{2}{e}^{2}+24\,xab{e}^{2}+6\,x{b}^{2}de+10\,{a}^{2}{e}^{2}+4\,abde+{b}^{2}{d}^{2}}{60\,{e}^{3} \left ( ex+d \right ) ^{6} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

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,x)

[Out]

-1/60/e^3*(15*b^2*e^2*x^2+24*a*b*e^2*x+6*b^2*d*e*x+10*a^2*e^2+4*a*b*d*e+b^2*d^2)
*((b*x+a)^2)^(1/2)/(e*x+d)^6/(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276305, size = 162, normalized size = 1.11 \[ -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(15*b^2*e^2*x^2 + b^2*d^2 + 4*a*b*d*e + 10*a^2*e^2 + 6*(b^2*d*e + 4*a*b*e^
2)*x)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^5*x^2
+ 6*d^5*e^4*x + d^6*e^3)

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Sympy [A]  time = 5.23898, size = 128, normalized size = 0.88 \[ - \frac{10 a^{2} e^{2} + 4 a b d e + b^{2} d^{2} + 15 b^{2} e^{2} x^{2} + x \left (24 a b e^{2} + 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]

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,x)

[Out]

-(10*a**2*e**2 + 4*a*b*d*e + b**2*d**2 + 15*b**2*e**2*x**2 + x*(24*a*b*e**2 + 6*
b**2*d*e))/(60*d**6*e**3 + 360*d**5*e**4*x + 900*d**4*e**5*x**2 + 1200*d**3*e**6
*x**3 + 900*d**2*e**7*x**4 + 360*d*e**8*x**5 + 60*e**9*x**6)

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GIAC/XCAS [A]  time = 0.285179, size = 130, normalized size = 0.89 \[ -\frac{{\left (15 \, b^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{2} d x e{\rm sign}\left (b x + a\right ) + b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 24 \, a b x e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a b d e{\rm sign}\left (b x + a\right ) + 10 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(15*b^2*x^2*e^2*sign(b*x + a) + 6*b^2*d*x*e*sign(b*x + a) + b^2*d^2*sign(b
*x + a) + 24*a*b*x*e^2*sign(b*x + a) + 4*a*b*d*e*sign(b*x + a) + 10*a^2*e^2*sign
(b*x + a))*e^(-3)/(x*e + d)^6