3.1397 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}}+\frac{3 d^3 \sqrt{c+d x}}{64 b^2 (a+b x) (b c-a d)^2}-\frac{d^2 \sqrt{c+d x}}{32 b^2 (a+b x)^2 (b c-a d)}-\frac{d \sqrt{c+d x}}{8 b^2 (a+b x)^3}-\frac{(c+d x)^{3/2}}{4 b (a+b x)^4} \]

[Out]

-(d*Sqrt[c + d*x])/(8*b^2*(a + b*x)^3) - (d^2*Sqrt[c + d*x])/(32*b^2*(b*c - a*d)
*(a + b*x)^2) + (3*d^3*Sqrt[c + d*x])/(64*b^2*(b*c - a*d)^2*(a + b*x)) - (c + d*
x)^(3/2)/(4*b*(a + b*x)^4) - (3*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a
*d]])/(64*b^(5/2)*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.212483, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}}+\frac{3 d^3 \sqrt{c+d x}}{64 b^2 (a+b x) (b c-a d)^2}-\frac{d^2 \sqrt{c+d x}}{32 b^2 (a+b x)^2 (b c-a d)}-\frac{d \sqrt{c+d x}}{8 b^2 (a+b x)^3}-\frac{(c+d x)^{3/2}}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^5,x]

[Out]

-(d*Sqrt[c + d*x])/(8*b^2*(a + b*x)^3) - (d^2*Sqrt[c + d*x])/(32*b^2*(b*c - a*d)
*(a + b*x)^2) + (3*d^3*Sqrt[c + d*x])/(64*b^2*(b*c - a*d)^2*(a + b*x)) - (c + d*
x)^(3/2)/(4*b*(a + b*x)^4) - (3*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a
*d]])/(64*b^(5/2)*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 40.1706, size = 150, normalized size = 0.87 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}}}{4 b \left (a + b x\right )^{4}} + \frac{3 d^{3} \sqrt{c + d x}}{64 b^{2} \left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{d^{2} \sqrt{c + d x}}{32 b^{2} \left (a + b x\right )^{2} \left (a d - b c\right )} - \frac{d \sqrt{c + d x}}{8 b^{2} \left (a + b x\right )^{3}} + \frac{3 d^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{64 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**5,x)

[Out]

-(c + d*x)**(3/2)/(4*b*(a + b*x)**4) + 3*d**3*sqrt(c + d*x)/(64*b**2*(a + b*x)*(
a*d - b*c)**2) + d**2*sqrt(c + d*x)/(32*b**2*(a + b*x)**2*(a*d - b*c)) - d*sqrt(
c + d*x)/(8*b**2*(a + b*x)**3) + 3*d**4*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*
c))/(64*b**(5/2)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 0.243253, size = 149, normalized size = 0.87 \[ -\frac{3 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x} \left (2 d^2 (a+b x)^2 (b c-a d)+24 d (a+b x) (b c-a d)^2+16 (b c-a d)^3-3 d^3 (a+b x)^3\right )}{64 b^2 (a+b x)^4 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^5,x]

[Out]

-(Sqrt[c + d*x]*(16*(b*c - a*d)^3 + 24*d*(b*c - a*d)^2*(a + b*x) + 2*d^2*(b*c -
a*d)*(a + b*x)^2 - 3*d^3*(a + b*x)^3))/(64*b^2*(b*c - a*d)^2*(a + b*x)^4) - (3*d
^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(64*b^(5/2)*(b*c - a*d)^(5/
2))

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Maple [A]  time = 0.023, size = 222, normalized size = 1.3 \[{\frac{3\,{d}^{4}b}{64\, \left ( bdx+ad \right ) ^{4} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{d}^{4}}{64\, \left ( bdx+ad \right ) ^{4} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{11\,{d}^{4}}{64\, \left ( bdx+ad \right ) ^{4}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{5}a}{64\, \left ( bdx+ad \right ) ^{4}{b}^{2}}\sqrt{dx+c}}+{\frac{3\,{d}^{4}c}{64\, \left ( bdx+ad \right ) ^{4}b}\sqrt{dx+c}}+{\frac{3\,{d}^{4}}{ \left ( 64\,{a}^{2}{d}^{2}-128\,abcd+64\,{b}^{2}{c}^{2} \right ){b}^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^5,x)

[Out]

3/64*d^4/(b*d*x+a*d)^4*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(7/2)+11/64*d^4/(b*
d*x+a*d)^4/(a*d-b*c)*(d*x+c)^(5/2)-11/64*d^4/(b*d*x+a*d)^4/b*(d*x+c)^(3/2)-3/64*
d^5/(b*d*x+a*d)^4/b^2*(d*x+c)^(1/2)*a+3/64*d^4/(b*d*x+a*d)^4/b*(d*x+c)^(1/2)*c+3
/64*d^4/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)
*b/((a*d-b*c)*b)^(1/2))

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240447, 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)^5,x, algorithm="fricas")

[Out]

[1/128*(2*(3*b^3*d^3*x^3 - 16*b^3*c^3 + 24*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 3*a^3*d
^3 - (2*b^3*c*d^2 - 11*a*b^2*d^3)*x^2 - (24*b^3*c^2*d - 44*a*b^2*c*d^2 + 11*a^2*
b*d^3)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c) + 3*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 +
6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*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^4*b^4*c^2 - 2*a^
5*b^3*c*d + a^6*b^2*d^2 + (b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*x^4 + 4*(a*b^7*c
^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*x^3 + 6*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4
*d^2)*x^2 + 4*(a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*x)*sqrt(b^2*c - a*b*d)
), 1/64*((3*b^3*d^3*x^3 - 16*b^3*c^3 + 24*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 3*a^3*d^
3 - (2*b^3*c*d^2 - 11*a*b^2*d^3)*x^2 - (24*b^3*c^2*d - 44*a*b^2*c*d^2 + 11*a^2*b
*d^3)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c) - 3*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 +
6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*arctan(-(b*c - a*d)/(sqrt(-b^2*c +
a*b*d)*sqrt(d*x + c))))/((a^4*b^4*c^2 - 2*a^5*b^3*c*d + a^6*b^2*d^2 + (b^8*c^2
- 2*a*b^7*c*d + a^2*b^6*d^2)*x^4 + 4*(a*b^7*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*x
^3 + 6*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*x^2 + 4*(a^3*b^5*c^2 - 2*a^4*
b^4*c*d + a^5*b^3*d^2)*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((d*x+c)**(3/2)/(b*x+a)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233422, size = 385, normalized size = 2.24 \[ \frac{3 \, d^{4} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{64 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} d^{4} - 11 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c d^{4} - 11 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{2} d^{4} + 3 \, \sqrt{d x + c} b^{3} c^{3} d^{4} + 11 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} d^{5} + 22 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c d^{5} - 9 \, \sqrt{d x + c} a b^{2} c^{2} d^{5} - 11 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{6} + 9 \, \sqrt{d x + c} a^{2} b c d^{6} - 3 \, \sqrt{d x + c} a^{3} d^{7}}{64 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^5,x, algorithm="giac")

[Out]

3/64*d^4*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2)*sqrt(-b^2*c + a*b*d)) + 1/64*(3*(d*x + c)^(7/2)*b^3*d^4 - 11*(d*x +
c)^(5/2)*b^3*c*d^4 - 11*(d*x + c)^(3/2)*b^3*c^2*d^4 + 3*sqrt(d*x + c)*b^3*c^3*d
^4 + 11*(d*x + c)^(5/2)*a*b^2*d^5 + 22*(d*x + c)^(3/2)*a*b^2*c*d^5 - 9*sqrt(d*x
+ c)*a*b^2*c^2*d^5 - 11*(d*x + c)^(3/2)*a^2*b*d^6 + 9*sqrt(d*x + c)*a^2*b*c*d^6
- 3*sqrt(d*x + c)*a^3*d^7)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*((d*x + c)*b -
b*c + a*d)^4)