3.1423 \(\int \frac{1}{(a+b x)^5 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=180 \[ -\frac{35 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 \sqrt{b} (b c-a d)^{9/2}}+\frac{35 d^3 \sqrt{c+d x}}{64 (a+b x) (b c-a d)^4}-\frac{35 d^2 \sqrt{c+d x}}{96 (a+b x)^2 (b c-a d)^3}+\frac{7 d \sqrt{c+d x}}{24 (a+b x)^3 (b c-a d)^2}-\frac{\sqrt{c+d x}}{4 (a+b x)^4 (b c-a d)} \]

[Out]

-Sqrt[c + d*x]/(4*(b*c - a*d)*(a + b*x)^4) + (7*d*Sqrt[c + d*x])/(24*(b*c - a*d)
^2*(a + b*x)^3) - (35*d^2*Sqrt[c + d*x])/(96*(b*c - a*d)^3*(a + b*x)^2) + (35*d^
3*Sqrt[c + d*x])/(64*(b*c - a*d)^4*(a + b*x)) - (35*d^4*ArcTanh[(Sqrt[b]*Sqrt[c
+ d*x])/Sqrt[b*c - a*d]])/(64*Sqrt[b]*(b*c - a*d)^(9/2))

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Rubi [A]  time = 0.199234, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{35 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{64 \sqrt{b} (b c-a d)^{9/2}}+\frac{35 d^3 \sqrt{c+d x}}{64 (a+b x) (b c-a d)^4}-\frac{35 d^2 \sqrt{c+d x}}{96 (a+b x)^2 (b c-a d)^3}+\frac{7 d \sqrt{c+d x}}{24 (a+b x)^3 (b c-a d)^2}-\frac{\sqrt{c+d x}}{4 (a+b x)^4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^5*Sqrt[c + d*x]),x]

[Out]

-Sqrt[c + d*x]/(4*(b*c - a*d)*(a + b*x)^4) + (7*d*Sqrt[c + d*x])/(24*(b*c - a*d)
^2*(a + b*x)^3) - (35*d^2*Sqrt[c + d*x])/(96*(b*c - a*d)^3*(a + b*x)^2) + (35*d^
3*Sqrt[c + d*x])/(64*(b*c - a*d)^4*(a + b*x)) - (35*d^4*ArcTanh[(Sqrt[b]*Sqrt[c
+ d*x])/Sqrt[b*c - a*d]])/(64*Sqrt[b]*(b*c - a*d)^(9/2))

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Rubi in Sympy [A]  time = 44.4349, size = 158, normalized size = 0.88 \[ \frac{35 d^{3} \sqrt{c + d x}}{64 \left (a + b x\right ) \left (a d - b c\right )^{4}} + \frac{35 d^{2} \sqrt{c + d x}}{96 \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} + \frac{7 d \sqrt{c + d x}}{24 \left (a + b x\right )^{3} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x}}{4 \left (a + b x\right )^{4} \left (a d - b c\right )} + \frac{35 d^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{64 \sqrt{b} \left (a d - b c\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**5/(d*x+c)**(1/2),x)

[Out]

35*d**3*sqrt(c + d*x)/(64*(a + b*x)*(a*d - b*c)**4) + 35*d**2*sqrt(c + d*x)/(96*
(a + b*x)**2*(a*d - b*c)**3) + 7*d*sqrt(c + d*x)/(24*(a + b*x)**3*(a*d - b*c)**2
) + sqrt(c + d*x)/(4*(a + b*x)**4*(a*d - b*c)) + 35*d**4*atan(sqrt(b)*sqrt(c + d
*x)/sqrt(a*d - b*c))/(64*sqrt(b)*(a*d - b*c)**(9/2))

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Mathematica [A]  time = 0.460604, size = 145, normalized size = 0.81 \[ \frac{1}{192} \left (\frac{\sqrt{c+d x} \left (70 d^2 (a+b x)^2 (a d-b c)+56 d (a+b x) (b c-a d)^2-48 (b c-a d)^3+105 d^3 (a+b x)^3\right )}{(a+b x)^4 (b c-a d)^4}-\frac{105 d^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^5*Sqrt[c + d*x]),x]

[Out]

((Sqrt[c + d*x]*(-48*(b*c - a*d)^3 + 56*d*(b*c - a*d)^2*(a + b*x) + 70*d^2*(-(b*
c) + a*d)*(a + b*x)^2 + 105*d^3*(a + b*x)^3))/((b*c - a*d)^4*(a + b*x)^4) - (105
*d^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b*c - a*d)^(9/2
)))/192

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Maple [A]  time = 0.011, size = 179, normalized size = 1. \[{\frac{{d}^{4}}{ \left ( 4\,ad-4\,bc \right ) \left ( bdx+ad \right ) ^{4}}\sqrt{dx+c}}+{\frac{7\,{d}^{4}}{24\, \left ( ad-bc \right ) ^{2} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}+{\frac{35\,{d}^{4}}{96\, \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{35\,{d}^{4}}{64\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{35\,{d}^{4}}{64\, \left ( ad-bc \right ) ^{4}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^5/(d*x+c)^(1/2),x)

[Out]

1/4*d^4*(d*x+c)^(1/2)/(a*d-b*c)/(b*d*x+a*d)^4+7/24*d^4/(a*d-b*c)^2*(d*x+c)^(1/2)
/(b*d*x+a*d)^3+35/96*d^4/(a*d-b*c)^3*(d*x+c)^(1/2)/(b*d*x+a*d)^2+35/64*d^4/(a*d-
b*c)^4*(d*x+c)^(1/2)/(b*d*x+a*d)+35/64*d^4/(a*d-b*c)^4/((a*d-b*c)*b)^(1/2)*arcta
n((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(1/((b*x + a)^5*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226233, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^5*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

[1/384*(2*(105*b^3*d^3*x^3 - 48*b^3*c^3 + 200*a*b^2*c^2*d - 326*a^2*b*c*d^2 + 27
9*a^3*d^3 - 35*(2*b^3*c*d^2 - 11*a*b^2*d^3)*x^2 + 7*(8*b^3*c^2*d - 36*a*b^2*c*d^
2 + 73*a^2*b*d^3)*x)*sqrt(b^2*c - a*b*d)*sqrt(d*x + c) + 105*(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^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^
4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^4 + 4*(
a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)
*x^3 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 +
a^6*b^2*d^4)*x^2 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*
b^2*c*d^3 + a^7*b*d^4)*x)*sqrt(b^2*c - a*b*d)), 1/192*((105*b^3*d^3*x^3 - 48*b^3
*c^3 + 200*a*b^2*c^2*d - 326*a^2*b*c*d^2 + 279*a^3*d^3 - 35*(2*b^3*c*d^2 - 11*a*
b^2*d^3)*x^2 + 7*(8*b^3*c^2*d - 36*a*b^2*c*d^2 + 73*a^2*b*d^3)*x)*sqrt(-b^2*c +
a*b*d)*sqrt(d*x + c) - 105*(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^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^
4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4
)*x^4 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a
^5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b
^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d
^2 - 4*a^6*b^2*c*d^3 + a^7*b*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(1/(b*x+a)**5/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220294, size = 447, normalized size = 2.48 \[ \frac{35 \, d^{4} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{64 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} + \frac{105 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} d^{4} - 385 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c d^{4} + 511 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{2} d^{4} - 279 \, \sqrt{d x + c} b^{3} c^{3} d^{4} + 385 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} d^{5} - 1022 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c d^{5} + 837 \, \sqrt{d x + c} a b^{2} c^{2} d^{5} + 511 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{6} - 837 \, \sqrt{d x + c} a^{2} b c d^{6} + 279 \, \sqrt{d x + c} a^{3} d^{7}}{192 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^5*sqrt(d*x + c)),x, algorithm="giac")

[Out]

35/64*d^4*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^4 - 4*a*b^3*c^3*d
+ 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b^2*c + a*b*d)) + 1/192*(1
05*(d*x + c)^(7/2)*b^3*d^4 - 385*(d*x + c)^(5/2)*b^3*c*d^4 + 511*(d*x + c)^(3/2)
*b^3*c^2*d^4 - 279*sqrt(d*x + c)*b^3*c^3*d^4 + 385*(d*x + c)^(5/2)*a*b^2*d^5 - 1
022*(d*x + c)^(3/2)*a*b^2*c*d^5 + 837*sqrt(d*x + c)*a*b^2*c^2*d^5 + 511*(d*x + c
)^(3/2)*a^2*b*d^6 - 837*sqrt(d*x + c)*a^2*b*c*d^6 + 279*sqrt(d*x + c)*a^3*d^7)/(
(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*((d*x +
c)*b - b*c + a*d)^4)