3.1386 \(\int \frac{\sqrt{c+d x}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=218 \[ -\frac{7 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac{7 d^4 \sqrt{c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac{7 d^3 \sqrt{c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac{7 d^2 \sqrt{c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac{d \sqrt{c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x}}{5 b (a+b x)^5} \]

[Out]

-Sqrt[c + d*x]/(5*b*(a + b*x)^5) - (d*Sqrt[c + d*x])/(40*b*(b*c - a*d)*(a + b*x)
^4) + (7*d^2*Sqrt[c + d*x])/(240*b*(b*c - a*d)^2*(a + b*x)^3) - (7*d^3*Sqrt[c +
d*x])/(192*b*(b*c - a*d)^3*(a + b*x)^2) + (7*d^4*Sqrt[c + d*x])/(128*b*(b*c - a*
d)^4*(a + b*x)) - (7*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*
b^(3/2)*(b*c - a*d)^(9/2))

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Rubi [A]  time = 0.390767, antiderivative size = 218, 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 \[ -\frac{7 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac{7 d^4 \sqrt{c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac{7 d^3 \sqrt{c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac{7 d^2 \sqrt{c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac{d \sqrt{c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x]/(5*b*(a + b*x)^5) - (d*Sqrt[c + d*x])/(40*b*(b*c - a*d)*(a + b*x)
^4) + (7*d^2*Sqrt[c + d*x])/(240*b*(b*c - a*d)^2*(a + b*x)^3) - (7*d^3*Sqrt[c +
d*x])/(192*b*(b*c - a*d)^3*(a + b*x)^2) + (7*d^4*Sqrt[c + d*x])/(128*b*(b*c - a*
d)^4*(a + b*x)) - (7*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(128*
b^(3/2)*(b*c - a*d)^(9/2))

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Rubi in Sympy [A]  time = 58.6843, size = 187, normalized size = 0.86 \[ \frac{7 d^{4} \sqrt{c + d x}}{128 b \left (a + b x\right ) \left (a d - b c\right )^{4}} + \frac{7 d^{3} \sqrt{c + d x}}{192 b \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} + \frac{7 d^{2} \sqrt{c + d x}}{240 b \left (a + b x\right )^{3} \left (a d - b c\right )^{2}} + \frac{d \sqrt{c + d x}}{40 b \left (a + b x\right )^{4} \left (a d - b c\right )} - \frac{\sqrt{c + d x}}{5 b \left (a + b x\right )^{5}} + \frac{7 d^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{128 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2)/(b*x+a)**6,x)

[Out]

7*d**4*sqrt(c + d*x)/(128*b*(a + b*x)*(a*d - b*c)**4) + 7*d**3*sqrt(c + d*x)/(19
2*b*(a + b*x)**2*(a*d - b*c)**3) + 7*d**2*sqrt(c + d*x)/(240*b*(a + b*x)**3*(a*d
- b*c)**2) + d*sqrt(c + d*x)/(40*b*(a + b*x)**4*(a*d - b*c)) - sqrt(c + d*x)/(5
*b*(a + b*x)**5) + 7*d**5*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(128*b**(3
/2)*(a*d - b*c)**(9/2))

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Mathematica [A]  time = 0.347278, size = 171, normalized size = 0.78 \[ -\frac{7 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left (70 d^3 (a+b x)^3 (b c-a d)-56 d^2 (a+b x)^2 (b c-a d)^2+48 d (a+b x) (b c-a d)^3+384 (b c-a d)^4-105 d^4 (a+b x)^4\right )}{1920 b (a+b x)^5 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x]*(384*(b*c - a*d)^4 + 48*d*(b*c - a*d)^3*(a + b*x) - 56*d^2*(b*c
- a*d)^2*(a + b*x)^2 + 70*d^3*(b*c - a*d)*(a + b*x)^3 - 105*d^4*(a + b*x)^4))/(1
920*b*(b*c - a*d)^4*(a + b*x)^5) - (7*d^5*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b
*c - a*d]])/(128*b^(3/2)*(b*c - a*d)^(9/2))

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Maple [A]  time = 0.026, size = 337, normalized size = 1.6 \[{\frac{7\,{d}^{5}{b}^{3}}{128\, \left ( bdx+ad \right ) ^{5} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) } \left ( dx+c \right ) ^{{\frac{9}{2}}}}+{\frac{49\,{d}^{5}{b}^{2}}{192\, \left ( bdx+ad \right ) ^{5} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{d}^{5}b}{15\, \left ( bdx+ad \right ) ^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{79\,{d}^{5}}{192\, \left ( bdx+ad \right ) ^{5} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{d}^{5}}{128\, \left ( bdx+ad \right ) ^{5}b}\sqrt{dx+c}}+{\frac{7\,{d}^{5}}{128\,b \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/2)/(b*x+a)^6,x)

[Out]

7/128*d^5/(b*d*x+a*d)^5*b^3/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3
*d+b^4*c^4)*(d*x+c)^(9/2)+49/192*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)^(7/2)+7/15*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)^(5/2)+79/192*d^5/(b*d*x+a*d)^5/(a*d-b*c)*(d*x+c)^(3/2)-7/128*d^5
/(b*d*x+a*d)^5/b*(d*x+c)^(1/2)+7/128*d^5/b/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*
d^2-4*a*b^3*c^3*d+b^4*c^4)/((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(sqrt(d*x + c)/(b*x + a)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237692, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(105*b^4*d^4*x^4 - 384*b^4*c^4 + 1488*a*b^3*c^3*d - 2104*a^2*b^2*c^2*
d^2 + 1210*a^3*b*c*d^3 - 105*a^4*d^4 - 70*(b^4*c*d^3 - 7*a*b^3*d^4)*x^3 + 14*(4*
b^4*c^2*d^2 - 23*a*b^3*c*d^3 + 64*a^2*b^2*d^4)*x^2 - 2*(24*b^4*c^3*d - 128*a*b^3
*c^2*d^2 + 289*a^2*b^2*c*d^3 - 395*a^3*b*d^4)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x +
c) + 105*(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^4 - 4*a^6*b^4*c^3*d + 6*a
^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4 + (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2
*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3
*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^4 + 10*(a^2*b^8*c^4 -
4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^3 + 10*(a
^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4
)*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 +
a^8*b^2*d^4)*x)*sqrt(b^2*c - a*b*d)), 1/1920*((105*b^4*d^4*x^4 - 384*b^4*c^4 +
1488*a*b^3*c^3*d - 2104*a^2*b^2*c^2*d^2 + 1210*a^3*b*c*d^3 - 105*a^4*d^4 - 70*(b
^4*c*d^3 - 7*a*b^3*d^4)*x^3 + 14*(4*b^4*c^2*d^2 - 23*a*b^3*c*d^3 + 64*a^2*b^2*d^
4)*x^2 - 2*(24*b^4*c^3*d - 128*a*b^3*c^2*d^2 + 289*a^2*b^2*c*d^3 - 395*a^3*b*d^4
)*x)*sqrt(-b^2*c + a*b*d)*sqrt(d*x + c) - 105*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 1
0*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^4 - 4*a^6*b^4*c^3*d + 6
*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4 + (b^10*c^4 - 4*a*b^9*c^3*d + 6*a
^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c
^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^4 + 10*(a^2*b^8*c^4
- 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^3 + 10*
(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d
^4)*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3
+ a^8*b^2*d^4)*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)**(1/2)/(b*x+a)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228928, size = 583, normalized size = 2.67 \[ \frac{7 \, d^{5} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{128 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt{-b^{2} c + a b d}} + \frac{105 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{4} d^{5} - 490 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{4} c d^{5} + 896 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} c^{2} d^{5} - 790 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{3} d^{5} - 105 \, \sqrt{d x + c} b^{4} c^{4} d^{5} + 490 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{3} d^{6} - 1792 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{3} c d^{6} + 2370 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d^{6} + 420 \, \sqrt{d x + c} a b^{3} c^{3} d^{6} + 896 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{2} d^{7} - 2370 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{7} - 630 \, \sqrt{d x + c} a^{2} b^{2} c^{2} d^{7} + 790 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b d^{8} + 420 \, \sqrt{d x + c} a^{3} b c d^{8} - 105 \, \sqrt{d x + c} a^{4} d^{9}}{1920 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/128*d^5*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^4 - 4*a*b^4*c^3*d
+ 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*sqrt(-b^2*c + a*b*d)) + 1/19
20*(105*(d*x + c)^(9/2)*b^4*d^5 - 490*(d*x + c)^(7/2)*b^4*c*d^5 + 896*(d*x + c)^
(5/2)*b^4*c^2*d^5 - 790*(d*x + c)^(3/2)*b^4*c^3*d^5 - 105*sqrt(d*x + c)*b^4*c^4*
d^5 + 490*(d*x + c)^(7/2)*a*b^3*d^6 - 1792*(d*x + c)^(5/2)*a*b^3*c*d^6 + 2370*(d
*x + c)^(3/2)*a*b^3*c^2*d^6 + 420*sqrt(d*x + c)*a*b^3*c^3*d^6 + 896*(d*x + c)^(5
/2)*a^2*b^2*d^7 - 2370*(d*x + c)^(3/2)*a^2*b^2*c*d^7 - 630*sqrt(d*x + c)*a^2*b^2
*c^2*d^7 + 790*(d*x + c)^(3/2)*a^3*b*d^8 + 420*sqrt(d*x + c)*a^3*b*c*d^8 - 105*s
qrt(d*x + c)*a^4*d^9)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*
c*d^3 + a^4*b*d^4)*((d*x + c)*b - b*c + a*d)^5)