3.1410 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=198 \[ -\frac{3 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}+\frac{3 d^4 \sqrt{c+d x}}{128 b^3 (a+b x) (b c-a d)^2}-\frac{d^3 \sqrt{c+d x}}{64 b^3 (a+b x)^2 (b c-a d)}-\frac{d^2 \sqrt{c+d x}}{16 b^3 (a+b x)^3}-\frac{d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac{(c+d x)^{5/2}}{5 b (a+b x)^5} \]

[Out]

-(d^2*Sqrt[c + d*x])/(16*b^3*(a + b*x)^3) - (d^3*Sqrt[c + d*x])/(64*b^3*(b*c - a
*d)*(a + b*x)^2) + (3*d^4*Sqrt[c + d*x])/(128*b^3*(b*c - a*d)^2*(a + b*x)) - (d*
(c + d*x)^(3/2))/(8*b^2*(a + b*x)^4) - (c + d*x)^(5/2)/(5*b*(a + b*x)^5) - (3*d^
5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*b^(7/2)*(b*c - a*d)^(5/
2))

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Rubi [A]  time = 0.264509, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}+\frac{3 d^4 \sqrt{c+d x}}{128 b^3 (a+b x) (b c-a d)^2}-\frac{d^3 \sqrt{c+d x}}{64 b^3 (a+b x)^2 (b c-a d)}-\frac{d^2 \sqrt{c+d x}}{16 b^3 (a+b x)^3}-\frac{d (c+d x)^{3/2}}{8 b^2 (a+b x)^4}-\frac{(c+d x)^{5/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(a + b*x)^6,x]

[Out]

-(d^2*Sqrt[c + d*x])/(16*b^3*(a + b*x)^3) - (d^3*Sqrt[c + d*x])/(64*b^3*(b*c - a
*d)*(a + b*x)^2) + (3*d^4*Sqrt[c + d*x])/(128*b^3*(b*c - a*d)^2*(a + b*x)) - (d*
(c + d*x)^(3/2))/(8*b^2*(a + b*x)^4) - (c + d*x)^(5/2)/(5*b*(a + b*x)^5) - (3*d^
5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*b^(7/2)*(b*c - a*d)^(5/
2))

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Rubi in Sympy [A]  time = 53.9616, size = 173, normalized size = 0.87 \[ - \frac{\left (c + d x\right )^{\frac{5}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{d \left (c + d x\right )^{\frac{3}{2}}}{8 b^{2} \left (a + b x\right )^{4}} + \frac{3 d^{4} \sqrt{c + d x}}{128 b^{3} \left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{d^{3} \sqrt{c + d x}}{64 b^{3} \left (a + b x\right )^{2} \left (a d - b c\right )} - \frac{d^{2} \sqrt{c + d x}}{16 b^{3} \left (a + b x\right )^{3}} + \frac{3 d^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{128 b^{\frac{7}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/(b*x+a)**6,x)

[Out]

-(c + d*x)**(5/2)/(5*b*(a + b*x)**5) - d*(c + d*x)**(3/2)/(8*b**2*(a + b*x)**4)
+ 3*d**4*sqrt(c + d*x)/(128*b**3*(a + b*x)*(a*d - b*c)**2) + d**3*sqrt(c + d*x)/
(64*b**3*(a + b*x)**2*(a*d - b*c)) - d**2*sqrt(c + d*x)/(16*b**3*(a + b*x)**3) +
3*d**5*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(128*b**(7/2)*(a*d - b*c)**(
5/2))

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Mathematica [A]  time = 0.336709, size = 171, normalized size = 0.86 \[ -\frac{3 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{7/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x} \left (10 d^3 (a+b x)^3 (b c-a d)+248 d^2 (a+b x)^2 (b c-a d)^2+336 d (a+b x) (b c-a d)^3+128 (b c-a d)^4-15 d^4 (a+b x)^4\right )}{640 b^3 (a+b x)^5 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(a + b*x)^6,x]

[Out]

-(Sqrt[c + d*x]*(128*(b*c - a*d)^4 + 336*d*(b*c - a*d)^3*(a + b*x) + 248*d^2*(b*
c - a*d)^2*(a + b*x)^2 + 10*d^3*(b*c - a*d)*(a + b*x)^3 - 15*d^4*(a + b*x)^4))/(
640*b^3*(b*c - a*d)^2*(a + b*x)^5) - (3*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt
[b*c - a*d]])/(128*b^(7/2)*(b*c - a*d)^(5/2))

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Maple [A]  time = 0.026, size = 305, normalized size = 1.5 \[{\frac{3\,{d}^{5}b}{128\, \left ( bdx+ad \right ) ^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{9}{2}}}}+{\frac{7\,{d}^{5}}{64\, \left ( bdx+ad \right ) ^{5} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{{d}^{5}}{5\, \left ( bdx+ad \right ) ^{5}b} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{d}^{6}a}{64\, \left ( bdx+ad \right ) ^{5}{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{d}^{5}c}{64\, \left ( bdx+ad \right ) ^{5}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{7}{a}^{2}}{128\, \left ( bdx+ad \right ) ^{5}{b}^{3}}\sqrt{dx+c}}+{\frac{3\,{d}^{6}ac}{64\, \left ( bdx+ad \right ) ^{5}{b}^{2}}\sqrt{dx+c}}-{\frac{3\,{d}^{5}{c}^{2}}{128\, \left ( bdx+ad \right ) ^{5}b}\sqrt{dx+c}}+{\frac{3\,{d}^{5}}{128\,{b}^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^6,x)

[Out]

3/128*d^5/(b*d*x+a*d)^5*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x+c)^(9/2)+7/64*d^5/(b*
d*x+a*d)^5/(a*d-b*c)*(d*x+c)^(7/2)-1/5*d^5/(b*d*x+a*d)^5/b*(d*x+c)^(5/2)-7/64*d^
6/(b*d*x+a*d)^5/b^2*(d*x+c)^(3/2)*a+7/64*d^5/(b*d*x+a*d)^5/b*(d*x+c)^(3/2)*c-3/1
28*d^7/(b*d*x+a*d)^5/b^3*(d*x+c)^(1/2)*a^2+3/64*d^6/(b*d*x+a*d)^5/b^2*(d*x+c)^(1
/2)*a*c-3/128*d^5/(b*d*x+a*d)^5/b*(d*x+c)^(1/2)*c^2+3/128*d^5/b^3/(a^2*d^2-2*a*b
*c*d+b^2*c^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)^(5/2)/(b*x + a)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227272, 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)^(5/2)/(b*x + a)^6,x, algorithm="fricas")

[Out]

[1/1280*(2*(15*b^4*d^4*x^4 - 128*b^4*c^4 + 176*a*b^3*c^3*d - 8*a^2*b^2*c^2*d^2 -
10*a^3*b*c*d^3 - 15*a^4*d^4 - 10*(b^4*c*d^3 - 7*a*b^3*d^4)*x^3 - 2*(124*b^4*c^2
*d^2 - 233*a*b^3*c*d^3 + 64*a^2*b^2*d^4)*x^2 - 2*(168*b^4*c^3*d - 256*a*b^3*c^2*
d^2 + 23*a^2*b^2*c*d^3 + 35*a^3*b*d^4)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c) + 15
*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^
4*b*d^5*x + a^5*d^5)*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^5*c^2 - 2*a^6*b^4*c*d + a^7*b^3*d^2 +
(b^10*c^2 - 2*a*b^9*c*d + a^2*b^8*d^2)*x^5 + 5*(a*b^9*c^2 - 2*a^2*b^8*c*d + a^3
*b^7*d^2)*x^4 + 10*(a^2*b^8*c^2 - 2*a^3*b^7*c*d + a^4*b^6*d^2)*x^3 + 10*(a^3*b^7
*c^2 - 2*a^4*b^6*c*d + a^5*b^5*d^2)*x^2 + 5*(a^4*b^6*c^2 - 2*a^5*b^5*c*d + a^6*b
^4*d^2)*x)*sqrt(b^2*c - a*b*d)), 1/640*((15*b^4*d^4*x^4 - 128*b^4*c^4 + 176*a*b^
3*c^3*d - 8*a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 - 15*a^4*d^4 - 10*(b^4*c*d^3 - 7*a*
b^3*d^4)*x^3 - 2*(124*b^4*c^2*d^2 - 233*a*b^3*c*d^3 + 64*a^2*b^2*d^4)*x^2 - 2*(1
68*b^4*c^3*d - 256*a*b^3*c^2*d^2 + 23*a^2*b^2*c*d^3 + 35*a^3*b*d^4)*x)*sqrt(-b^2
*c + a*b*d)*sqrt(d*x + c) - 15*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x
^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*arctan(-(b*c - a*d)/(sqrt(-b^
2*c + a*b*d)*sqrt(d*x + c))))/((a^5*b^5*c^2 - 2*a^6*b^4*c*d + a^7*b^3*d^2 + (b^1
0*c^2 - 2*a*b^9*c*d + a^2*b^8*d^2)*x^5 + 5*(a*b^9*c^2 - 2*a^2*b^8*c*d + a^3*b^7*
d^2)*x^4 + 10*(a^2*b^8*c^2 - 2*a^3*b^7*c*d + a^4*b^6*d^2)*x^3 + 10*(a^3*b^7*c^2
- 2*a^4*b^6*c*d + a^5*b^5*d^2)*x^2 + 5*(a^4*b^6*c^2 - 2*a^5*b^5*c*d + a^6*b^4*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)**(5/2)/(b*x+a)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.241769, size = 513, normalized size = 2.59 \[ \frac{3 \, d^{5} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{128 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{15 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{4} d^{5} - 70 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{4} c d^{5} - 128 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} c^{2} d^{5} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{3} d^{5} - 15 \, \sqrt{d x + c} b^{4} c^{4} d^{5} + 70 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{3} d^{6} + 256 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{3} c d^{6} - 210 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d^{6} + 60 \, \sqrt{d x + c} a b^{3} c^{3} d^{6} - 128 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{2} d^{7} + 210 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{7} - 90 \, \sqrt{d x + c} a^{2} b^{2} c^{2} d^{7} - 70 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b d^{8} + 60 \, \sqrt{d x + c} a^{3} b c d^{8} - 15 \, \sqrt{d x + c} a^{4} d^{9}}{640 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^6,x, algorithm="giac")

[Out]

3/128*d^5*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^2 - 2*a*b^4*c*d +
a^2*b^3*d^2)*sqrt(-b^2*c + a*b*d)) + 1/640*(15*(d*x + c)^(9/2)*b^4*d^5 - 70*(d*
x + c)^(7/2)*b^4*c*d^5 - 128*(d*x + c)^(5/2)*b^4*c^2*d^5 + 70*(d*x + c)^(3/2)*b^
4*c^3*d^5 - 15*sqrt(d*x + c)*b^4*c^4*d^5 + 70*(d*x + c)^(7/2)*a*b^3*d^6 + 256*(d
*x + c)^(5/2)*a*b^3*c*d^6 - 210*(d*x + c)^(3/2)*a*b^3*c^2*d^6 + 60*sqrt(d*x + c)
*a*b^3*c^3*d^6 - 128*(d*x + c)^(5/2)*a^2*b^2*d^7 + 210*(d*x + c)^(3/2)*a^2*b^2*c
*d^7 - 90*sqrt(d*x + c)*a^2*b^2*c^2*d^7 - 70*(d*x + c)^(3/2)*a^3*b*d^8 + 60*sqrt
(d*x + c)*a^3*b*c*d^8 - 15*sqrt(d*x + c)*a^4*d^9)/((b^5*c^2 - 2*a*b^4*c*d + a^2*
b^3*d^2)*((d*x + c)*b - b*c + a*d)^5)