3.1759 \(\int \frac{(a+b x)^2 \sqrt{e+f x}}{c+d x} \, dx\)

Optimal. Leaf size=138 \[ -\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 \sqrt{e+f x} (b c-a d)^2}{d^3}-\frac{2 b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{3 d^2 f^2}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2} \]

[Out]

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

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Rubi [A]  time = 0.253491, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 \sqrt{e+f x} (b c-a d)^2}{d^3}-\frac{2 b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{3 d^2 f^2}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.9727, size = 129, normalized size = 0.93 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{5}{2}}}{5 d f^{2}} + \frac{2 b \left (e + f x\right )^{\frac{3}{2}} \left (2 a d f - b c f - b d e\right )}{3 d^{2} f^{2}} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{2}}{d^{3}} - \frac{2 \left (a d - b c\right )^{2} \sqrt{c f - d e} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.203872, size = 152, normalized size = 1.1 \[ \frac{2 \sqrt{e+f x} \left (15 a^2 d^2 f^2+10 a b d f (d (e+f x)-3 c f)+b^2 \left (15 c^2 f^2-5 c d f (e+f x)+d^2 \left (-2 e^2+e f x+3 f^2 x^2\right )\right )\right )}{15 d^3 f^2}-\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e + f*x]*(15*a^2*d^2*f^2 + 10*a*b*d*f*(-3*c*f + d*(e + f*x)) + b^2*(15*c
^2*f^2 - 5*c*d*f*(e + f*x) + d^2*(-2*e^2 + e*f*x + 3*f^2*x^2))))/(15*d^3*f^2) -
(2*(b*c - a*d)^2*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]
])/d^(7/2)

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Maple [B]  time = 0.017, size = 387, normalized size = 2.8 \[{\frac{2\,{b}^{2}}{5\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+{\frac{4\,ab}{3\,df} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{b}^{2}c}{3\,f{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{b}^{2}e}{3\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}\sqrt{fx+e}}{d}}-4\,{\frac{abc\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{{b}^{2}{c}^{2}\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{{a}^{2}fc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{a}^{2}e}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{bfa{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abce}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{2}{c}^{3}f}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}e}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b^2*(f*x+e)^(5/2)/d/f^2+4/3/f/d*(f*x+e)^(3/2)*a*b-2/3/f/d^2*(f*x+e)^(3/2)*b^
2*c-2/3/f^2/d*(f*x+e)^(3/2)*b^2*e+2/d*a^2*(f*x+e)^(1/2)-4/d^2*a*b*c*(f*x+e)^(1/2
)+2/d^3*b^2*c^2*(f*x+e)^(1/2)-2*f/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/(
(c*f-d*e)*d)^(1/2))*a^2*c+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e
)*d)^(1/2))*a^2*e+4*f/d^2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*
d)^(1/2))*a*b*c^2-4/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(
1/2))*a*b*c*e-2*f/d^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^3+2/d^2/((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*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((b*x + a)^2*sqrt(f*x + e)/(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224154, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) + 2 \,{\left (3 \, b^{2} d^{2} f^{2} x^{2} - 2 \, b^{2} d^{2} e^{2} - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} e f + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} +{\left (b^{2} d^{2} e f - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}}{15 \, d^{3} f^{2}}, -\frac{2 \,{\left (15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) -{\left (3 \, b^{2} d^{2} f^{2} x^{2} - 2 \, b^{2} d^{2} e^{2} - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} e f + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} +{\left (b^{2} d^{2} e f - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{15 \, d^{3} f^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*sqrt((d*e - c*f)/d)*log((d*f*x + 2
*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) + 2*(3*b^2*d^2*f^
2*x^2 - 2*b^2*d^2*e^2 - 5*(b^2*c*d - 2*a*b*d^2)*e*f + 15*(b^2*c^2 - 2*a*b*c*d +
a^2*d^2)*f^2 + (b^2*d^2*e*f - 5*(b^2*c*d - 2*a*b*d^2)*f^2)*x)*sqrt(f*x + e))/(d^
3*f^2), -2/15*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*sqrt(-(d*e - c*f)/d)*arcta
n(sqrt(f*x + e)/sqrt(-(d*e - c*f)/d)) - (3*b^2*d^2*f^2*x^2 - 2*b^2*d^2*e^2 - 5*(
b^2*c*d - 2*a*b*d^2)*e*f + 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2 + (b^2*d^2*e*f
- 5*(b^2*c*d - 2*a*b*d^2)*f^2)*x)*sqrt(f*x + e))/(d^3*f^2)]

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Sympy [A]  time = 13.9607, size = 274, normalized size = 1.99 \[ \frac{2 \left (\frac{b^{2} \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a b d f - b^{2} c f - b^{2} d e\right )}{3 d^{2} f} - \frac{f \left (a d - b c\right )^{2} \left (c f - d e\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{3}} + \frac{\sqrt{e + f x} \left (a^{2} d^{2} f - 2 a b c d f + b^{2} c^{2} f\right )}{d^{3}}\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b**2*(e + f*x)**(5/2)/(5*d*f) + (e + f*x)**(3/2)*(2*a*b*d*f - b**2*c*f - b**2
*d*e)/(3*d**2*f) - f*(a*d - b*c)**2*(c*f - d*e)*Piecewise((atan(sqrt(e + f*x)/sq
rt((c*f - d*e)/d))/(d*sqrt((c*f - d*e)/d)), (c*f - d*e)/d > 0), (-acoth(sqrt(e +
f*x)/sqrt((-c*f + d*e)/d))/(d*sqrt((-c*f + d*e)/d)), ((c*f - d*e)/d < 0) & (e +
f*x > (-c*f + d*e)/d)), (-atanh(sqrt(e + f*x)/sqrt((-c*f + d*e)/d))/(d*sqrt((-c
*f + d*e)/d)), ((c*f - d*e)/d < 0) & (e + f*x < (-c*f + d*e)/d)))/d**3 + sqrt(e
+ f*x)*(a**2*d**2*f - 2*a*b*c*d*f + b**2*c**2*f)/d**3)/f

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GIAC/XCAS [A]  time = 0.217278, size = 338, normalized size = 2.45 \[ -\frac{2 \,{\left (b^{2} c^{3} f - 2 \, a b c^{2} d f + a^{2} c d^{2} f - b^{2} c^{2} d e + 2 \, a b c d^{2} e - a^{2} d^{3} e\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{2} d^{4} f^{8} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c d^{3} f^{9} + 10 \,{\left (f x + e\right )}^{\frac{3}{2}} a b d^{4} f^{9} + 15 \, \sqrt{f x + e} b^{2} c^{2} d^{2} f^{10} - 30 \, \sqrt{f x + e} a b c d^{3} f^{10} + 15 \, \sqrt{f x + e} a^{2} d^{4} f^{10} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} d^{4} f^{8} e\right )}}{15 \, d^{5} f^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b^2*c^3*f - 2*a*b*c^2*d*f + a^2*c*d^2*f - b^2*c^2*d*e + 2*a*b*c*d^2*e - a^2*
d^3*e)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^3) + 2
/15*(3*(f*x + e)^(5/2)*b^2*d^4*f^8 - 5*(f*x + e)^(3/2)*b^2*c*d^3*f^9 + 10*(f*x +
e)^(3/2)*a*b*d^4*f^9 + 15*sqrt(f*x + e)*b^2*c^2*d^2*f^10 - 30*sqrt(f*x + e)*a*b
*c*d^3*f^10 + 15*sqrt(f*x + e)*a^2*d^4*f^10 - 5*(f*x + e)^(3/2)*b^2*d^4*f^8*e)/(
d^5*f^10)