3.1398 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=208 \[ \frac{3 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}}-\frac{3 d^4 \sqrt{c+d x}}{128 b^2 (a+b x) (b c-a d)^3}+\frac{d^3 \sqrt{c+d x}}{64 b^2 (a+b x)^2 (b c-a d)^2}-\frac{d^2 \sqrt{c+d x}}{80 b^2 (a+b x)^3 (b c-a d)}-\frac{3 d \sqrt{c+d x}}{40 b^2 (a+b x)^4}-\frac{(c+d x)^{3/2}}{5 b (a+b x)^5} \]

[Out]

(-3*d*Sqrt[c + d*x])/(40*b^2*(a + b*x)^4) - (d^2*Sqrt[c + d*x])/(80*b^2*(b*c - a
*d)*(a + b*x)^3) + (d^3*Sqrt[c + d*x])/(64*b^2*(b*c - a*d)^2*(a + b*x)^2) - (3*d
^4*Sqrt[c + d*x])/(128*b^2*(b*c - a*d)^3*(a + b*x)) - (c + d*x)^(3/2)/(5*b*(a +
b*x)^5) + (3*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*b^(5/2)*
(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.272974, antiderivative size = 208, 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 \[ \frac{3 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}}-\frac{3 d^4 \sqrt{c+d x}}{128 b^2 (a+b x) (b c-a d)^3}+\frac{d^3 \sqrt{c+d x}}{64 b^2 (a+b x)^2 (b c-a d)^2}-\frac{d^2 \sqrt{c+d x}}{80 b^2 (a+b x)^3 (b c-a d)}-\frac{3 d \sqrt{c+d x}}{40 b^2 (a+b x)^4}-\frac{(c+d x)^{3/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^6,x]

[Out]

(-3*d*Sqrt[c + d*x])/(40*b^2*(a + b*x)^4) - (d^2*Sqrt[c + d*x])/(80*b^2*(b*c - a
*d)*(a + b*x)^3) + (d^3*Sqrt[c + d*x])/(64*b^2*(b*c - a*d)^2*(a + b*x)^2) - (3*d
^4*Sqrt[c + d*x])/(128*b^2*(b*c - a*d)^3*(a + b*x)) - (c + d*x)^(3/2)/(5*b*(a +
b*x)^5) + (3*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*b^(5/2)*
(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 53.8691, size = 184, normalized size = 0.88 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}}}{5 b \left (a + b x\right )^{5}} + \frac{3 d^{4} \sqrt{c + d x}}{128 b^{2} \left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{d^{3} \sqrt{c + d x}}{64 b^{2} \left (a + b x\right )^{2} \left (a d - b c\right )^{2}} + \frac{d^{2} \sqrt{c + d x}}{80 b^{2} \left (a + b x\right )^{3} \left (a d - b c\right )} - \frac{3 d \sqrt{c + d x}}{40 b^{2} \left (a + b x\right )^{4}} + \frac{3 d^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{128 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**6,x)

[Out]

-(c + d*x)**(3/2)/(5*b*(a + b*x)**5) + 3*d**4*sqrt(c + d*x)/(128*b**2*(a + b*x)*
(a*d - b*c)**3) + d**3*sqrt(c + d*x)/(64*b**2*(a + b*x)**2*(a*d - b*c)**2) + d**
2*sqrt(c + d*x)/(80*b**2*(a + b*x)**3*(a*d - b*c)) - 3*d*sqrt(c + d*x)/(40*b**2*
(a + b*x)**4) + 3*d**5*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(128*b**(5/2)
*(a*d - b*c)**(7/2))

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Mathematica [A]  time = 0.50343, size = 171, normalized size = 0.82 \[ \frac{3 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}}-\frac{\sqrt{c+d x} \left (10 d^3 (a+b x)^3 (a d-b c)+8 d^2 (a+b x)^2 (b c-a d)^2+176 d (a+b x) (b c-a d)^3+128 (b c-a d)^4+15 d^4 (a+b x)^4\right )}{640 b^2 (a+b x)^5 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^6,x]

[Out]

-(Sqrt[c + d*x]*(128*(b*c - a*d)^4 + 176*d*(b*c - a*d)^3*(a + b*x) + 8*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^2*(b*c - a*d)^3*(a + b*x)^5) + (3*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqr
t[b*c - a*d]])/(128*b^(5/2)*(b*c - a*d)^(7/2))

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Maple [A]  time = 0.024, size = 300, normalized size = 1.4 \[{\frac{3\,{d}^{5}{b}^{2}}{128\, \left ( bdx+ad \right ) ^{5} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) } \left ( dx+c \right ) ^{{\frac{9}{2}}}}+{\frac{7\,{d}^{5}b}{64\, \left ( bdx+ad \right ) ^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{5}}{5\, \left ( bdx+ad \right ) ^{5} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{d}^{5}}{64\, \left ( bdx+ad \right ) ^{5}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{6}a}{128\, \left ( bdx+ad \right ) ^{5}{b}^{2}}\sqrt{dx+c}}+{\frac{3\,{d}^{5}c}{128\, \left ( bdx+ad \right ) ^{5}b}\sqrt{dx+c}}+{\frac{3\,{d}^{5}}{ \left ( 128\,{a}^{3}{d}^{3}-384\,{a}^{2}bc{d}^{2}+384\,a{b}^{2}{c}^{2}d-128\,{b}^{3}{c}^{3} \right ){b}^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^6,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243816, 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)^(3/2)/(b*x + a)^6,x, algorithm="fricas")

[Out]

[-1/1280*(2*(15*b^4*d^4*x^4 + 128*b^4*c^4 - 336*a*b^3*c^3*d + 248*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*(4*b^4*c^
2*d^2 - 23*a*b^3*c*d^3 + 64*a^2*b^2*d^4)*x^2 + 2*(88*b^4*c^3*d - 256*a*b^3*c^2*d
^2 + 233*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^3 - 3*a^6*b^4*c^2*d + 3*a^7*b^3*c
*d^2 - a^8*b^2*d^3 + (b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*
x^5 + 5*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*x^4 + 10*(
a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*x^3 + 10*(a^3*b^7
*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*x^2 + 5*(a^4*b^6*c^3 - 3
*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*x)*sqrt(b^2*c - a*b*d)), -1/640*
((15*b^4*d^4*x^4 + 128*b^4*c^4 - 336*a*b^3*c^3*d + 248*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*(4*b^4*c^2*d^2 - 23*
a*b^3*c*d^3 + 64*a^2*b^2*d^4)*x^2 + 2*(88*b^4*c^3*d - 256*a*b^3*c^2*d^2 + 233*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^3 - 3*a^6*b^4*c^2*d + 3*a^7*b^3*c*d^2 - a^8*b^2*d^3 + (b^10*c^3 - 3*a*b^9*c^2*
d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*x^5 + 5*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*
b^7*c*d^2 - a^4*b^6*d^3)*x^4 + 10*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d
^2 - a^5*b^5*d^3)*x^3 + 10*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^
6*b^4*d^3)*x^2 + 5*(a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^
3)*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)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233644, size = 554, normalized size = 2.66 \[ -\frac{3 \, d^{5} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{128 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{15 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{4} d^{5} - 70 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{4} c d^{5} + 128 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} c^{2} d^{5} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{3} d^{5} - 15 \, \sqrt{d x + c} b^{4} c^{4} d^{5} + 70 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{3} d^{6} - 256 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{3} c d^{6} - 210 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d^{6} + 60 \, \sqrt{d x + c} a b^{3} c^{3} d^{6} + 128 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{2} d^{7} + 210 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{7} - 90 \, \sqrt{d x + c} a^{2} b^{2} c^{2} d^{7} - 70 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b d^{8} + 60 \, \sqrt{d x + c} a^{3} b c d^{8} - 15 \, \sqrt{d x + c} a^{4} d^{9}}{640 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(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^3 - 3*a*b^4*c^2*
d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*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^3
- 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*((d*x + c)*b - b*c + a*d)^5)