3.1982 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=98 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{30 (d+e x)^5 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{6 (d+e x)^6 (b d-a e)} \]

[Out]

((a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(b*d - a*e)*(d + e*x)^6) + (b*(a
+ b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(30*(b*d - a*e)^2*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.168744, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{30 (d+e x)^5 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{6 (d+e x)^6 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

((a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(b*d - a*e)*(d + e*x)^6) + (b*(a
+ b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(30*(b*d - a*e)^2*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.9177, size = 73, normalized size = 0.74 \[ \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{30 \left (d + e x\right )^{5} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{6 \left (d + e x\right )^{6} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**7,x)

[Out]

b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(30*(d + e*x)**5*(a*e - b*d)**2) - (a**2 +
2*a*b*x + b**2*x**2)**(5/2)/(6*(d + e*x)**6*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.11475, size = 162, normalized size = 1.65 \[ -\frac{\sqrt{(a+b x)^2} \left (5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{30 e^5 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(5*a^4*e^4 + 4*a^3*b*e^3*(d + 6*e*x) + 3*a^2*b^2*e^2*(d^2 +
6*d*e*x + 15*e^2*x^2) + 2*a*b^3*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)
+ b^4*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)))/(30*e^5*(
a + b*x)*(d + e*x)^6)

_______________________________________________________________________________________

Maple [B]  time = 0.012, size = 201, normalized size = 2.1 \[ -{\frac{15\,{x}^{4}{b}^{4}{e}^{4}+40\,{x}^{3}a{b}^{3}{e}^{4}+20\,{x}^{3}{b}^{4}d{e}^{3}+45\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+30\,{x}^{2}a{b}^{3}d{e}^{3}+15\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+24\,x{a}^{3}b{e}^{4}+18\,x{a}^{2}{b}^{2}d{e}^{3}+12\,xa{b}^{3}{d}^{2}{e}^{2}+6\,x{b}^{4}{d}^{3}e+5\,{a}^{4}{e}^{4}+4\,{a}^{3}bd{e}^{3}+3\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+2\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{30\,{e}^{5} \left ( ex+d \right ) ^{6} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x)

[Out]

-1/30/e^5*(15*b^4*e^4*x^4+40*a*b^3*e^4*x^3+20*b^4*d*e^3*x^3+45*a^2*b^2*e^4*x^2+3
0*a*b^3*d*e^3*x^2+15*b^4*d^2*e^2*x^2+24*a^3*b*e^4*x+18*a^2*b^2*d*e^3*x+12*a*b^3*
d^2*e^2*x+6*b^4*d^3*e*x+5*a^4*e^4+4*a^3*b*d*e^3+3*a^2*b^2*d^2*e^2+2*a*b^3*d^3*e+
b^4*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^6/(b*x+a)^3

_______________________________________________________________________________________

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((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.281491, size = 319, normalized size = 3.26 \[ -\frac{15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \,{\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^7,x, algorithm="fricas")

[Out]

-1/30*(15*b^4*e^4*x^4 + b^4*d^4 + 2*a*b^3*d^3*e + 3*a^2*b^2*d^2*e^2 + 4*a^3*b*d*
e^3 + 5*a^4*e^4 + 20*(b^4*d*e^3 + 2*a*b^3*e^4)*x^3 + 15*(b^4*d^2*e^2 + 2*a*b^3*d
*e^3 + 3*a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 2*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 + 4
*a^3*b*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d
^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)

_______________________________________________________________________________________

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**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278197, size = 356, normalized size = 3.63 \[ -\frac{{\left (15 \, b^{4} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 20 \, b^{4} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \, b^{4} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{4} d^{3} x e{\rm sign}\left (b x + a\right ) + b^{4} d^{4}{\rm sign}\left (b x + a\right ) + 40 \, a b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 30 \, a b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 12 \, a b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 45 \, a^{2} b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 18 \, a^{2} b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 24 \, a^{3} b x e^{4}{\rm sign}\left (b x + a\right ) + 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)/(e*x + d)^7,x, algorithm="giac")

[Out]

-1/30*(15*b^4*x^4*e^4*sign(b*x + a) + 20*b^4*d*x^3*e^3*sign(b*x + a) + 15*b^4*d^
2*x^2*e^2*sign(b*x + a) + 6*b^4*d^3*x*e*sign(b*x + a) + b^4*d^4*sign(b*x + a) +
40*a*b^3*x^3*e^4*sign(b*x + a) + 30*a*b^3*d*x^2*e^3*sign(b*x + a) + 12*a*b^3*d^2
*x*e^2*sign(b*x + a) + 2*a*b^3*d^3*e*sign(b*x + a) + 45*a^2*b^2*x^2*e^4*sign(b*x
+ a) + 18*a^2*b^2*d*x*e^3*sign(b*x + a) + 3*a^2*b^2*d^2*e^2*sign(b*x + a) + 24*
a^3*b*x*e^4*sign(b*x + a) + 4*a^3*b*d*e^3*sign(b*x + a) + 5*a^4*e^4*sign(b*x + a
))*e^(-5)/(x*e + d)^6