3.450 \(\int \frac{(c+d x)^{5/2}}{x^4 (a+b x)} \, dx\)

Optimal. Leaf size=207 \[ -\frac{2 \sqrt{b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^4}+\frac{c \sqrt{c+d x} (2 b c-3 a d)}{4 a^2 x^2}-\frac{\sqrt{c+d x} \left (11 a^2 d^2-18 a b c d+8 b^2 c^2\right )}{8 a^3 x}+\frac{\left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^4 \sqrt{c}}-\frac{c (c+d x)^{3/2}}{3 a x^3} \]

[Out]

(c*(2*b*c - 3*a*d)*Sqrt[c + d*x])/(4*a^2*x^2) - ((8*b^2*c^2 - 18*a*b*c*d + 11*a^
2*d^2)*Sqrt[c + d*x])/(8*a^3*x) - (c*(c + d*x)^(3/2))/(3*a*x^3) + ((16*b^3*c^3 -
40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(8
*a^4*Sqrt[c]) - (2*Sqrt[b]*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqr
t[b*c - a*d]])/a^4

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Rubi [A]  time = 0.7999, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{2 \sqrt{b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^4}+\frac{c \sqrt{c+d x} (2 b c-3 a d)}{4 a^2 x^2}-\frac{\sqrt{c+d x} \left (11 a^2 d^2-18 a b c d+8 b^2 c^2\right )}{8 a^3 x}+\frac{\left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{8 a^4 \sqrt{c}}-\frac{c (c+d x)^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^4*(a + b*x)),x]

[Out]

(c*(2*b*c - 3*a*d)*Sqrt[c + d*x])/(4*a^2*x^2) - ((8*b^2*c^2 - 18*a*b*c*d + 11*a^
2*d^2)*Sqrt[c + d*x])/(8*a^3*x) - (c*(c + d*x)^(3/2))/(3*a*x^3) + ((16*b^3*c^3 -
40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(8
*a^4*Sqrt[c]) - (2*Sqrt[b]*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqr
t[b*c - a*d]])/a^4

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Rubi in Sympy [A]  time = 88.9877, size = 199, normalized size = 0.96 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3 a x^{3}} - \frac{c \sqrt{c + d x} \left (3 a d - 2 b c\right )}{4 a^{2} x^{2}} - \frac{\sqrt{c + d x} \left (11 a^{2} d^{2} - 18 a b c d + 8 b^{2} c^{2}\right )}{8 a^{3} x} - \frac{2 \sqrt{b} \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{4}} - \frac{\left (5 a^{3} d^{3} - 30 a^{2} b c d^{2} + 40 a b^{2} c^{2} d - 16 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{8 a^{4} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**4/(b*x+a),x)

[Out]

-c*(c + d*x)**(3/2)/(3*a*x**3) - c*sqrt(c + d*x)*(3*a*d - 2*b*c)/(4*a**2*x**2) -
sqrt(c + d*x)*(11*a**2*d**2 - 18*a*b*c*d + 8*b**2*c**2)/(8*a**3*x) - 2*sqrt(b)*
(a*d - b*c)**(5/2)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/a**4 - (5*a**3*d*
*3 - 30*a**2*b*c*d**2 + 40*a*b**2*c**2*d - 16*b**3*c**3)*atanh(sqrt(c + d*x)/sqr
t(c))/(8*a**4*sqrt(c))

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Mathematica [A]  time = 0.314543, size = 178, normalized size = 0.86 \[ -\frac{\frac{a \sqrt{c+d x} \left (a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )-6 a b c x (2 c+9 d x)+24 b^2 c^2 x^2\right )}{x^3}-\frac{3 \left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+48 \sqrt{b} (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{24 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)),x]

[Out]

-((a*Sqrt[c + d*x]*(24*b^2*c^2*x^2 - 6*a*b*c*x*(2*c + 9*d*x) + a^2*(8*c^2 + 26*c
*d*x + 33*d^2*x^2)))/x^3 - (3*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*
a^3*d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/Sqrt[c] + 48*Sqrt[b]*(b*c - a*d)^(5/2)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(24*a^4)

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Maple [B]  time = 0.025, size = 461, normalized size = 2.2 \[ -{\frac{11}{8\,a{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{9\,bc}{4\,d{a}^{2}{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}{c}^{2}}{{d}^{2}{a}^{3}{x}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,c}{3\,a{x}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-4\,{\frac{ \left ( dx+c \right ) ^{3/2}b{c}^{2}}{d{a}^{2}{x}^{3}}}+2\,{\frac{ \left ( dx+c \right ) ^{3/2}{b}^{2}{c}^{3}}{{d}^{2}{a}^{3}{x}^{3}}}+{\frac{7\,{c}^{3}b}{4\,d{a}^{2}{x}^{3}}\sqrt{dx+c}}-{\frac{{b}^{2}{c}^{4}}{{d}^{2}{a}^{3}{x}^{3}}\sqrt{dx+c}}-{\frac{5\,{c}^{2}}{8\,a{x}^{3}}\sqrt{dx+c}}-{\frac{5\,{d}^{3}}{8\,a}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{15\,{d}^{2}b}{4\,{a}^{2}}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }-5\,{\frac{d{c}^{3/2}{b}^{2}}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{c}^{5/2}{b}^{3}}{{a}^{4}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{{d}^{3}b}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{d}^{2}{b}^{2}c}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{d{b}^{3}{c}^{2}}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{b}^{4}{c}^{3}}{{a}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^4/(b*x+a),x)

[Out]

-11/8/a/x^3*(d*x+c)^(5/2)+9/4/d/a^2/x^3*(d*x+c)^(5/2)*c*b-1/d^2/a^3/x^3*(d*x+c)^
(5/2)*b^2*c^2+5/3*c*(d*x+c)^(3/2)/a/x^3-4/d/a^2/x^3*(d*x+c)^(3/2)*b*c^2+2/d^2/a^
3/x^3*(d*x+c)^(3/2)*b^2*c^3+7/4/d/a^2/x^3*(d*x+c)^(1/2)*b*c^3-1/d^2/a^3/x^3*(d*x
+c)^(1/2)*b^2*c^4-5/8/a/x^3*(d*x+c)^(1/2)*c^2-5/8*d^3/a/c^(1/2)*arctanh((d*x+c)^
(1/2)/c^(1/2))+15/4*d^2/a^2*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b-5*d/a^3*c^(
3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b^2+2/a^4*c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1
/2))*b^3-2*d^3*b/a/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2
))+6*d^2*b^2/a^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))
*c-6*d*b^3/a^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c
^2+2*b^4/a^4/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^3

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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)^(5/2)/((b*x + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.580519, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)*x^4),x, algorithm="fricas")

[Out]

[1/48*(48*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(b^2*c - a*b*d)*sqrt(c)*x^3*log((b
*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 3*(16*b^3
*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*x^3*log(((d*x + 2*c)*sqrt(c)
- 2*sqrt(d*x + c)*c)/x) - 2*(8*a^3*c^2 + 3*(8*a*b^2*c^2 - 18*a^2*b*c*d + 11*a^3
*d^2)*x^2 - 2*(6*a^2*b*c^2 - 13*a^3*c*d)*x)*sqrt(d*x + c)*sqrt(c))/(a^4*sqrt(c)*
x^3), 1/48*(96*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-b^2*c + a*b*d)*sqrt(c)*x^3*
arctan(sqrt(-b^2*c + a*b*d)/(sqrt(d*x + c)*b)) - 3*(16*b^3*c^3 - 40*a*b^2*c^2*d
+ 30*a^2*b*c*d^2 - 5*a^3*d^3)*x^3*log(((d*x + 2*c)*sqrt(c) - 2*sqrt(d*x + c)*c)/
x) - 2*(8*a^3*c^2 + 3*(8*a*b^2*c^2 - 18*a^2*b*c*d + 11*a^3*d^2)*x^2 - 2*(6*a^2*b
*c^2 - 13*a^3*c*d)*x)*sqrt(d*x + c)*sqrt(c))/(a^4*sqrt(c)*x^3), 1/24*(24*(b^2*c^
2 - 2*a*b*c*d + a^2*d^2)*sqrt(b^2*c - a*b*d)*sqrt(-c)*x^3*log((b*d*x + 2*b*c - a
*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 3*(16*b^3*c^3 - 40*a*b^2*
c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*x^3*arctan(c/(sqrt(d*x + c)*sqrt(-c))) - (8*
a^3*c^2 + 3*(8*a*b^2*c^2 - 18*a^2*b*c*d + 11*a^3*d^2)*x^2 - 2*(6*a^2*b*c^2 - 13*
a^3*c*d)*x)*sqrt(d*x + c)*sqrt(-c))/(a^4*sqrt(-c)*x^3), 1/24*(48*(b^2*c^2 - 2*a*
b*c*d + a^2*d^2)*sqrt(-b^2*c + a*b*d)*sqrt(-c)*x^3*arctan(sqrt(-b^2*c + a*b*d)/(
sqrt(d*x + c)*b)) - 3*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)
*x^3*arctan(c/(sqrt(d*x + c)*sqrt(-c))) - (8*a^3*c^2 + 3*(8*a*b^2*c^2 - 18*a^2*b
*c*d + 11*a^3*d^2)*x^2 - 2*(6*a^2*b*c^2 - 13*a^3*c*d)*x)*sqrt(d*x + c)*sqrt(-c))
/(a^4*sqrt(-c)*x^3)]

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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)**(5/2)/x**4/(b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22632, size = 405, normalized size = 1.96 \[ \frac{2 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{4}} - \frac{{\left (16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{8 \, a^{4} \sqrt{-c}} - \frac{24 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} c^{2} d - 48 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c^{3} d + 24 \, \sqrt{d x + c} b^{2} c^{4} d - 54 \,{\left (d x + c\right )}^{\frac{5}{2}} a b c d^{2} + 96 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c^{2} d^{2} - 42 \, \sqrt{d x + c} a b c^{3} d^{2} + 33 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} d^{3} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} c d^{3} + 15 \, \sqrt{d x + c} a^{2} c^{2} d^{3}}{24 \, a^{3} d^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)*x^4),x, algorithm="giac")

[Out]

2*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*arctan(sqrt(d*x + c)*b
/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^4) - 1/8*(16*b^3*c^3 - 40*a*b^2*c
^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^4*sqrt(-c))
- 1/24*(24*(d*x + c)^(5/2)*b^2*c^2*d - 48*(d*x + c)^(3/2)*b^2*c^3*d + 24*sqrt(d
*x + c)*b^2*c^4*d - 54*(d*x + c)^(5/2)*a*b*c*d^2 + 96*(d*x + c)^(3/2)*a*b*c^2*d^
2 - 42*sqrt(d*x + c)*a*b*c^3*d^2 + 33*(d*x + c)^(5/2)*a^2*d^3 - 40*(d*x + c)^(3/
2)*a^2*c*d^3 + 15*sqrt(d*x + c)*a^2*c^2*d^3)/(a^3*d^3*x^3)