3.1763 \(\int \frac{(a+b x)^2}{(c+d x) (e+f x)^{7/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 (b e-a f) (a d f-2 b c f+b d e)}{3 f^2 (e+f x)^{3/2} (d e-c f)^2}+\frac{2 (b e-a f)^2}{5 f^2 (e+f x)^{5/2} (d e-c f)}+\frac{2 (b c-a d)^2}{\sqrt{e+f x} (d e-c f)^3}-\frac{2 \sqrt{d} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{7/2}} \]

[Out]

(2*(b*e - a*f)^2)/(5*f^2*(d*e - c*f)*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(b*d*e -
2*b*c*f + a*d*f))/(3*f^2*(d*e - c*f)^2*(e + f*x)^(3/2)) + (2*(b*c - a*d)^2)/((d*
e - c*f)^3*Sqrt[e + f*x]) - (2*Sqrt[d]*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[e + f
*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.459008, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 (b e-a f) (a d f-2 b c f+b d e)}{3 f^2 (e+f x)^{3/2} (d e-c f)^2}+\frac{2 (b e-a f)^2}{5 f^2 (e+f x)^{5/2} (d e-c f)}+\frac{2 (b c-a d)^2}{\sqrt{e+f x} (d e-c f)^3}-\frac{2 \sqrt{d} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*e - a*f)^2)/(5*f^2*(d*e - c*f)*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(b*d*e -
2*b*c*f + a*d*f))/(3*f^2*(d*e - c*f)^2*(e + f*x)^(3/2)) + (2*(b*c - a*d)^2)/((d*
e - c*f)^3*Sqrt[e + f*x]) - (2*Sqrt[d]*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[e + f
*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 87.6304, size = 155, normalized size = 0.9 \[ - \frac{2 \sqrt{d} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{\left (c f - d e\right )^{\frac{7}{2}}} - \frac{2 \left (a d - b c\right )^{2}}{\sqrt{e + f x} \left (c f - d e\right )^{3}} + \frac{2 \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{3 f^{2} \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{2}}{5 f^{2} \left (e + f x\right )^{\frac{5}{2}} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(d)*(a*d - b*c)**2*atan(sqrt(d)*sqrt(e + f*x)/sqrt(c*f - d*e))/(c*f - d*e
)**(7/2) - 2*(a*d - b*c)**2/(sqrt(e + f*x)*(c*f - d*e)**3) + 2*(a*f - b*e)*(a*d*
f - 2*b*c*f + b*d*e)/(3*f**2*(e + f*x)**(3/2)*(c*f - d*e)**2) - 2*(a*f - b*e)**2
/(5*f**2*(e + f*x)**(5/2)*(c*f - d*e))

_______________________________________________________________________________________

Mathematica [A]  time = 0.494044, size = 166, normalized size = 0.96 \[ \frac{2 \left (15 f^2 (e+f x)^2 (b c-a d)^2-5 (e+f x) (b e-a f) (d e-c f) (a d f-2 b c f+b d e)+3 (b e-a f)^2 (d e-c f)^2\right )}{15 f^2 (e+f x)^{5/2} (d e-c f)^3}-\frac{2 \sqrt{d} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(3*(b*e - a*f)^2*(d*e - c*f)^2 - 5*(b*e - a*f)*(d*e - c*f)*(b*d*e - 2*b*c*f +
a*d*f)*(e + f*x) + 15*(b*c - a*d)^2*f^2*(e + f*x)^2))/(15*f^2*(d*e - c*f)^3*(e
+ f*x)^(5/2)) - (2*Sqrt[d]*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*
e - c*f]])/(d*e - c*f)^(7/2)

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 408, normalized size = 2.4 \[ -{\frac{2\,{a}^{2}}{5\,cf-5\,de} \left ( fx+e \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,abe}{5\, \left ( cf-de \right ) f} \left ( fx+e \right ) ^{-{\frac{5}{2}}}}-{\frac{2\,{b}^{2}{e}^{2}}{5\,{f}^{2} \left ( cf-de \right ) } \left ( fx+e \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,{a}^{2}d}{3\, \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,abc}{3\, \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,ce{b}^{2}}{3\,f \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{b}^{2}d{e}^{2}}{3\,{f}^{2} \left ( cf-de \right ) ^{2}} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{{a}^{2}{d}^{2}}{ \left ( cf-de \right ) ^{3}\sqrt{fx+e}}}+4\,{\frac{abcd}{ \left ( cf-de \right ) ^{3}\sqrt{fx+e}}}-2\,{\frac{{b}^{2}{c}^{2}}{ \left ( cf-de \right ) ^{3}\sqrt{fx+e}}}-2\,{\frac{{d}^{3}{a}^{2}}{ \left ( cf-de \right ) ^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{{d}^{2}abc}{ \left ( cf-de \right ) ^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{2}d{c}^{2}}{ \left ( cf-de \right ) ^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5/(c*f-d*e)/(f*x+e)^(5/2)*a^2+4/5/f/(c*f-d*e)/(f*x+e)^(5/2)*a*b*e-2/5/f^2/(c*
f-d*e)/(f*x+e)^(5/2)*b^2*e^2+2/3/(c*f-d*e)^2/(f*x+e)^(3/2)*a^2*d-4/3/(c*f-d*e)^2
/(f*x+e)^(3/2)*a*b*c+4/3/f/(c*f-d*e)^2/(f*x+e)^(3/2)*b^2*c*e-2/3/f^2/(c*f-d*e)^2
/(f*x+e)^(3/2)*b^2*d*e^2-2/(c*f-d*e)^3/(f*x+e)^(1/2)*a^2*d^2+4/(c*f-d*e)^3/(f*x+
e)^(1/2)*a*b*c*d-2/(c*f-d*e)^3/(f*x+e)^(1/2)*b^2*c^2-2*d^3/(c*f-d*e)^3/((c*f-d*e
)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2+4*d^2/(c*f-d*e)^3/((c
*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a*b*c-2*d/(c*f-d*e)
^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*b^2*c^2

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231224, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(4*b^2*d^2*e^4 - 6*a^2*c^2*f^4 - 30*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^4*x
^2 - 6*(3*b^2*c*d - 2*a*b*d^2)*e^3*f - 2*(8*b^2*c^2 - 28*a*b*c*d + 23*a^2*d^2)*e
^2*f^2 - 2*(4*a*b*c^2 - 11*a^2*c*d)*e*f^3 + 15*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*
f^4*x^2 + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e*f^3*x + (b^2*c^2 - 2*a*b*c*d + a^2
*d^2)*e^2*f^2)*sqrt(f*x + e)*sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e - c*f + 2*(d
*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x + c)) + 10*(b^2*d^2*e^3*f - 3*
b^2*c*d*e^2*f^2 - (4*b^2*c^2 - 14*a*b*c*d + 7*a^2*d^2)*e*f^3 - (2*a*b*c^2 - a^2*
c*d)*f^4)*x)/((d^3*e^5*f^2 - 3*c*d^2*e^4*f^3 + 3*c^2*d*e^3*f^4 - c^3*e^2*f^5 + (
d^3*e^3*f^4 - 3*c*d^2*e^2*f^5 + 3*c^2*d*e*f^6 - c^3*f^7)*x^2 + 2*(d^3*e^4*f^3 -
3*c*d^2*e^3*f^4 + 3*c^2*d*e^2*f^5 - c^3*e*f^6)*x)*sqrt(f*x + e)), -2/15*(2*b^2*d
^2*e^4 - 3*a^2*c^2*f^4 - 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^4*x^2 - 3*(3*b^2*c
*d - 2*a*b*d^2)*e^3*f - (8*b^2*c^2 - 28*a*b*c*d + 23*a^2*d^2)*e^2*f^2 - (4*a*b*c
^2 - 11*a^2*c*d)*e*f^3 + 15*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^4*x^2 + 2*(b^2*c^
2 - 2*a*b*c*d + a^2*d^2)*e*f^3*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2*f^2)*sqrt
(f*x + e)*sqrt(-d/(d*e - c*f))*arctan(-(d*e - c*f)*sqrt(-d/(d*e - c*f))/(sqrt(f*
x + e)*d)) + 5*(b^2*d^2*e^3*f - 3*b^2*c*d*e^2*f^2 - (4*b^2*c^2 - 14*a*b*c*d + 7*
a^2*d^2)*e*f^3 - (2*a*b*c^2 - a^2*c*d)*f^4)*x)/((d^3*e^5*f^2 - 3*c*d^2*e^4*f^3 +
3*c^2*d*e^3*f^4 - c^3*e^2*f^5 + (d^3*e^3*f^4 - 3*c*d^2*e^2*f^5 + 3*c^2*d*e*f^6
- c^3*f^7)*x^2 + 2*(d^3*e^4*f^3 - 3*c*d^2*e^3*f^4 + 3*c^2*d*e^2*f^5 - c^3*e*f^6)
*x)*sqrt(f*x + e))]

_______________________________________________________________________________________

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)**2/(d*x+c)/(f*x+e)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224861, size = 583, normalized size = 3.37 \[ -\frac{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} \sqrt{c d f - d^{2} e}} - \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{2} b^{2} c^{2} f^{2} - 30 \,{\left (f x + e\right )}^{2} a b c d f^{2} + 15 \,{\left (f x + e\right )}^{2} a^{2} d^{2} f^{2} + 10 \,{\left (f x + e\right )} a b c^{2} f^{3} - 5 \,{\left (f x + e\right )} a^{2} c d f^{3} + 3 \, a^{2} c^{2} f^{4} - 10 \,{\left (f x + e\right )} b^{2} c^{2} f^{2} e - 10 \,{\left (f x + e\right )} a b c d f^{2} e + 5 \,{\left (f x + e\right )} a^{2} d^{2} f^{2} e - 6 \, a b c^{2} f^{3} e - 6 \, a^{2} c d f^{3} e + 15 \,{\left (f x + e\right )} b^{2} c d f e^{2} + 3 \, b^{2} c^{2} f^{2} e^{2} + 12 \, a b c d f^{2} e^{2} + 3 \, a^{2} d^{2} f^{2} e^{2} - 5 \,{\left (f x + e\right )} b^{2} d^{2} e^{3} - 6 \, b^{2} c d f e^{3} - 6 \, a b d^{2} f e^{3} + 3 \, b^{2} d^{2} e^{4}\right )}}{15 \,{\left (c^{3} f^{5} - 3 \, c^{2} d f^{4} e + 3 \, c d^{2} f^{3} e^{2} - d^{3} f^{2} e^{3}\right )}{\left (f x + e\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e
))/((c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^3*e^3)*sqrt(c*d*f - d^2*e)) - 2
/15*(15*(f*x + e)^2*b^2*c^2*f^2 - 30*(f*x + e)^2*a*b*c*d*f^2 + 15*(f*x + e)^2*a^
2*d^2*f^2 + 10*(f*x + e)*a*b*c^2*f^3 - 5*(f*x + e)*a^2*c*d*f^3 + 3*a^2*c^2*f^4 -
10*(f*x + e)*b^2*c^2*f^2*e - 10*(f*x + e)*a*b*c*d*f^2*e + 5*(f*x + e)*a^2*d^2*f
^2*e - 6*a*b*c^2*f^3*e - 6*a^2*c*d*f^3*e + 15*(f*x + e)*b^2*c*d*f*e^2 + 3*b^2*c^
2*f^2*e^2 + 12*a*b*c*d*f^2*e^2 + 3*a^2*d^2*f^2*e^2 - 5*(f*x + e)*b^2*d^2*e^3 - 6
*b^2*c*d*f*e^3 - 6*a*b*d^2*f*e^3 + 3*b^2*d^2*e^4)/((c^3*f^5 - 3*c^2*d*f^4*e + 3*
c*d^2*f^3*e^2 - d^3*f^2*e^3)*(f*x + e)^(5/2))