3.1700 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx\)

Optimal. Leaf size=170 \[ -\frac{\sqrt{b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac{\sqrt{c+d x} (b c-a d)}{b (a+b x) (b e-a f)}+\frac{2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{f} (b e-a f)^2} \]

[Out]

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

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Rubi [A]  time = 0.616161, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac{\sqrt{c+d x} (b c-a d)}{b (a+b x) (b e-a f)}+\frac{2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{f} (b e-a f)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 82.7584, size = 146, normalized size = 0.86 \[ - \frac{2 \left (c f - d e\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}}{\sqrt{f} \left (a f - b e\right )^{2}} - \frac{\sqrt{c + d x} \left (a d - b c\right )}{b \left (a + b x\right ) \left (a f - b e\right )} + \frac{\sqrt{a d - b c} \left (a d f + 2 b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \left (a f - b e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.428368, size = 156, normalized size = 0.92 \[ \frac{\frac{\sqrt{b c-a d} (a d f+2 b c f-3 b d e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}}+\frac{\sqrt{c+d x} (a d-b c) (b e-a f)}{b (a+b x)}-\frac{2 (c f-d e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )}{\sqrt{f}}}{(b e-a f)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.037, size = 549, normalized size = 3.2 \[ -2\,{\frac{{c}^{2}{f}^{2}}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( cf-de \right ) f}}{\it Artanh} \left ({\frac{\sqrt{dx+c}f}{\sqrt{ \left ( cf-de \right ) f}}} \right ) }+4\,{\frac{cdef}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( cf-de \right ) f}}{\it Artanh} \left ({\frac{\sqrt{dx+c}f}{\sqrt{ \left ( cf-de \right ) f}}} \right ) }-2\,{\frac{{d}^{2}{e}^{2}}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( cf-de \right ) f}}{\it Artanh} \left ({\frac{\sqrt{dx+c}f}{\sqrt{ \left ( cf-de \right ) f}}} \right ) }-{\frac{{a}^{2}{d}^{2}f}{ \left ( af-be \right ) ^{2}b \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{acdf}{ \left ( af-be \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{a{d}^{2}e}{ \left ( af-be \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-{\frac{bcde}{ \left ( af-be \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{{a}^{2}{d}^{2}f}{ \left ( af-be \right ) ^{2}b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{acdf}{ \left ( af-be \right ) ^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-2\,{\frac{b{c}^{2}f}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-3\,{\frac{a{d}^{2}e}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+3\,{\frac{bcde}{ \left ( af-be \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.820608, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*s
qrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)
/b))/(b*x + a)) + 2*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)*x)*sqrt(-(d*e - c*f
)/f)*log((d*f*x - d*e + 2*c*f - 2*sqrt(d*x + c)*f*sqrt(-(d*e - c*f)/f))/(f*x + e
)) + 2*((b^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2
*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f^2)*x), -((3*a*b*d*e -
(2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt(-(b*c - a*d)/b)*
arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + (a*b*d*e - a*b*c*f + (b^2*d*e - b^2
*c*f)*x)*sqrt(-(d*e - c*f)/f)*log((d*f*x - d*e + 2*c*f - 2*sqrt(d*x + c)*f*sqrt(
-(d*e - c*f)/f))/(f*x + e)) + ((b^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x +
c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f
^2)*x), 1/2*(4*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)*x)*sqrt((d*e - c*f)/f)*a
rctan(sqrt(d*x + c)/sqrt((d*e - c*f)/f)) - (3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3
*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d
+ 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) - 2*((b^2*c - a*b*d)*e - (a*
b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2
- 2*a*b^3*e*f + a^2*b^2*f^2)*x), -((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*
e - (2*b^2*c + a*b*d)*f)*x)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c
- a*d)/b)) - 2*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)*x)*sqrt((d*e - c*f)/f)*
arctan(sqrt(d*x + c)/sqrt((d*e - c*f)/f)) + ((b^2*c - a*b*d)*e - (a*b*c - a^2*d)
*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*e
*f + a^2*b^2*f^2)*x)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219433, size = 338, normalized size = 1.99 \[ -\frac{{\left (2 \, b^{2} c^{2} f - a b c d f - a^{2} d^{2} f - 3 \, b^{2} c d e + 3 \, a b d^{2} e\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b f^{2} - 2 \, a b^{2} f e + b^{3} e^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} f}{\sqrt{-c f^{2} + d f e}}\right )}{{\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )} \sqrt{-c f^{2} + d f e}} + \frac{\sqrt{d x + c} b c d - \sqrt{d x + c} a d^{2}}{{\left (a b f - b^{2} e\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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