3.14 \(\int \frac{\sqrt{c+d x} (e+f x)}{x (a+b x)^3} \, dx\)

Optimal. Leaf size=208 \[ -\frac{2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}+\frac{\sqrt{c+d x} \left (a^2 (-d) f-3 a b d e+4 b^2 c e\right )}{4 a^2 b (a+b x) (b c-a d)}+\frac{\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x} (b e-a f)}{2 a b (a+b x)^2} \]

[Out]

((b*e - a*f)*Sqrt[c + d*x])/(2*a*b*(a + b*x)^2) + ((4*b^2*c*e - 3*a*b*d*e - a^2*
d*f)*Sqrt[c + d*x])/(4*a^2*b*(b*c - a*d)*(a + b*x)) - (2*Sqrt[c]*e*ArcTanh[Sqrt[
c + d*x]/Sqrt[c]])/a^3 + ((8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e + a^3*d^
2*f)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*a^3*b^(3/2)*(b*c - a*d
)^(3/2))

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Rubi [A]  time = 0.833057, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}+\frac{\sqrt{c+d x} \left (a^2 (-d) f-3 a b d e+4 b^2 c e\right )}{4 a^2 b (a+b x) (b c-a d)}+\frac{\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x} (b e-a f)}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*e - a*f)*Sqrt[c + d*x])/(2*a*b*(a + b*x)^2) + ((4*b^2*c*e - 3*a*b*d*e - a^2*
d*f)*Sqrt[c + d*x])/(4*a^2*b*(b*c - a*d)*(a + b*x)) - (2*Sqrt[c]*e*ArcTanh[Sqrt[
c + d*x]/Sqrt[c]])/a^3 + ((8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e + a^3*d^
2*f)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*a^3*b^(3/2)*(b*c - a*d
)^(3/2))

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Rubi in Sympy [A]  time = 86.9637, size = 189, normalized size = 0.91 \[ - \frac{\sqrt{c + d x} \left (a f - b e\right )}{2 a b \left (a + b x\right )^{2}} + \frac{\sqrt{c + d x} \left (a d \left (a f + 3 b e\right ) - 4 b^{2} c e\right )}{4 a^{2} b \left (a + b x\right ) \left (a d - b c\right )} - \frac{2 \sqrt{c} e \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3}} + \frac{\left (a^{3} d^{2} f + 3 a^{2} b d^{2} e - 12 a b^{2} c d e + 8 b^{3} c^{2} e\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 a^{3} b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a)**3,x)

[Out]

-sqrt(c + d*x)*(a*f - b*e)/(2*a*b*(a + b*x)**2) + sqrt(c + d*x)*(a*d*(a*f + 3*b*
e) - 4*b**2*c*e)/(4*a**2*b*(a + b*x)*(a*d - b*c)) - 2*sqrt(c)*e*atanh(sqrt(c + d
*x)/sqrt(c))/a**3 + (a**3*d**2*f + 3*a**2*b*d**2*e - 12*a*b**2*c*d*e + 8*b**3*c*
*2*e)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(4*a**3*b**(3/2)*(a*d - b*c)**
(3/2))

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Mathematica [A]  time = 0.571909, size = 184, normalized size = 0.88 \[ \frac{\frac{a \sqrt{c+d x} \left (\frac{(a+b x) \left (a^2 d f+3 a b d e-4 b^2 c e\right )}{a d-b c}+2 a (b e-a f)\right )}{b (a+b x)^2}+\frac{\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{3/2}}-8 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((a*Sqrt[c + d*x]*(2*a*(b*e - a*f) + ((-4*b^2*c*e + 3*a*b*d*e + a^2*d*f)*(a + b*
x))/(-(b*c) + a*d)))/(b*(a + b*x)^2) - 8*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]
] + ((8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e + a^3*d^2*f)*ArcTanh[(Sqrt[b]
*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(3/2)))/(4*a^3)

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Maple [B]  time = 0.028, size = 424, normalized size = 2. \[ -2\,{\frac{e\sqrt{c}}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{{d}^{2}f}{4\, \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}be}{4\,a \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}cde}{{a}^{2} \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{2}f}{4\, \left ( bdx+ad \right ) ^{2}b}\sqrt{dx+c}}+{\frac{5\,{d}^{2}e}{4\,a \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{bdce}{{a}^{2} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{{d}^{2}f}{ \left ( 4\,ad-4\,bc \right ) b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{3\,{d}^{2}e}{4\,a \left ( ad-bc \right ) }\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-3\,{\frac{bdce}{{a}^{2} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}e}{{a}^{3} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.839792, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(8*((b^4*c - a*b^3*d)*e*x^2 + 2*(a*b^3*c - a^2*b^2*d)*e*x + (a^2*b^2*c - a^
3*b*d)*e)*sqrt(b^2*c - a*b*d)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/
x) + 2*sqrt(b^2*c - a*b*d)*((6*a^2*b^2*c - 5*a^3*b*d)*e - (2*a^3*b*c - a^4*d)*f
- (a^3*b*d*f - (4*a*b^3*c - 3*a^2*b^2*d)*e)*x)*sqrt(d*x + c) - (a^5*d^2*f + (a^3
*b^2*d^2*f + (8*b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c^2
- 12*a^3*b^2*c*d + 3*a^4*b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c
*d + 3*a^3*b^2*d^2)*e)*x)*log((sqrt(b^2*c - a*b*d)*(b*d*x + 2*b*c - a*d) - 2*(b^
2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)))/((a^5*b^2*c - a^6*b*d + (a^3*b^4*c - a^4
*b^3*d)*x^2 + 2*(a^4*b^3*c - a^5*b^2*d)*x)*sqrt(b^2*c - a*b*d)), 1/4*(4*((b^4*c
- a*b^3*d)*e*x^2 + 2*(a*b^3*c - a^2*b^2*d)*e*x + (a^2*b^2*c - a^3*b*d)*e)*sqrt(-
b^2*c + a*b*d)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + sqrt(-b^2*
c + a*b*d)*((6*a^2*b^2*c - 5*a^3*b*d)*e - (2*a^3*b*c - a^4*d)*f - (a^3*b*d*f - (
4*a*b^3*c - 3*a^2*b^2*d)*e)*x)*sqrt(d*x + c) + (a^5*d^2*f + (a^3*b^2*d^2*f + (8*
b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c^2 - 12*a^3*b^2*c*d
+ 3*a^4*b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c*d + 3*a^3*b^2*d
^2)*e)*x)*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))))/((a^5*b^2*c
- a^6*b*d + (a^3*b^4*c - a^4*b^3*d)*x^2 + 2*(a^4*b^3*c - a^5*b^2*d)*x)*sqrt(-b^
2*c + a*b*d)), -1/8*(16*((b^4*c - a*b^3*d)*e*x^2 + 2*(a*b^3*c - a^2*b^2*d)*e*x +
(a^2*b^2*c - a^3*b*d)*e)*sqrt(b^2*c - a*b*d)*sqrt(-c)*arctan(sqrt(d*x + c)/sqrt
(-c)) - 2*sqrt(b^2*c - a*b*d)*((6*a^2*b^2*c - 5*a^3*b*d)*e - (2*a^3*b*c - a^4*d)
*f - (a^3*b*d*f - (4*a*b^3*c - 3*a^2*b^2*d)*e)*x)*sqrt(d*x + c) + (a^5*d^2*f + (
a^3*b^2*d^2*f + (8*b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c
^2 - 12*a^3*b^2*c*d + 3*a^4*b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^
3*c*d + 3*a^3*b^2*d^2)*e)*x)*log((sqrt(b^2*c - a*b*d)*(b*d*x + 2*b*c - a*d) - 2*
(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)))/((a^5*b^2*c - a^6*b*d + (a^3*b^4*c -
a^4*b^3*d)*x^2 + 2*(a^4*b^3*c - a^5*b^2*d)*x)*sqrt(b^2*c - a*b*d)), -1/4*(8*((b^
4*c - a*b^3*d)*e*x^2 + 2*(a*b^3*c - a^2*b^2*d)*e*x + (a^2*b^2*c - a^3*b*d)*e)*sq
rt(-b^2*c + a*b*d)*sqrt(-c)*arctan(sqrt(d*x + c)/sqrt(-c)) - sqrt(-b^2*c + a*b*d
)*((6*a^2*b^2*c - 5*a^3*b*d)*e - (2*a^3*b*c - a^4*d)*f - (a^3*b*d*f - (4*a*b^3*c
- 3*a^2*b^2*d)*e)*x)*sqrt(d*x + c) - (a^5*d^2*f + (a^3*b^2*d^2*f + (8*b^5*c^2 -
12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c^2 - 12*a^3*b^2*c*d + 3*a^4*
b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c*d + 3*a^3*b^2*d^2)*e)*x)
*arctan(-(b*c - a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))))/((a^5*b^2*c - a^6*b*
d + (a^3*b^4*c - a^4*b^3*d)*x^2 + 2*(a^4*b^3*c - a^5*b^2*d)*x)*sqrt(-b^2*c + a*b
*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((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228061, size = 405, normalized size = 1.95 \[ -\frac{{\left (a^{3} d^{2} f + 8 \, b^{3} c^{2} e - 12 \, a b^{2} c d e + 3 \, a^{2} b d^{2} e\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \, c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{a^{3} \sqrt{-c}} - \frac{{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{2} f + \sqrt{d x + c} a^{2} b c d^{2} f - \sqrt{d x + c} a^{3} d^{3} f - 4 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d e + 4 \, \sqrt{d x + c} b^{3} c^{2} d e + 3 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{2} e - 9 \, \sqrt{d x + c} a b^{2} c d^{2} e + 5 \, \sqrt{d x + c} a^{2} b d^{3} e}{4 \,{\left (a^{2} b^{2} c - a^{3} b d\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(a^3*d^2*f + 8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e)*arctan(sqrt(d*x
+ c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^2*c - a^4*b*d)*sqrt(-b^2*c + a*b*d)) + 2*c*
arctan(sqrt(d*x + c)/sqrt(-c))*e/(a^3*sqrt(-c)) - 1/4*((d*x + c)^(3/2)*a^2*b*d^2
*f + sqrt(d*x + c)*a^2*b*c*d^2*f - sqrt(d*x + c)*a^3*d^3*f - 4*(d*x + c)^(3/2)*b
^3*c*d*e + 4*sqrt(d*x + c)*b^3*c^2*d*e + 3*(d*x + c)^(3/2)*a*b^2*d^2*e - 9*sqrt(
d*x + c)*a*b^2*c*d^2*e + 5*sqrt(d*x + c)*a^2*b*d^3*e)/((a^2*b^2*c - a^3*b*d)*((d
*x + c)*b - b*c + a*d)^2)