3.1433 \(\int \frac{1}{(a+b x)^4 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{35 d^3}{8 \sqrt{c+d x} (b c-a d)^4}+\frac{35 \sqrt{b} d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 (b c-a d)^{9/2}}-\frac{35 d^2}{24 (a+b x) \sqrt{c+d x} (b c-a d)^3}+\frac{7 d}{12 (a+b x)^2 \sqrt{c+d x} (b c-a d)^2}-\frac{1}{3 (a+b x)^3 \sqrt{c+d x} (b c-a d)} \]

[Out]

(-35*d^3)/(8*(b*c - a*d)^4*Sqrt[c + d*x]) - 1/(3*(b*c - a*d)*(a + b*x)^3*Sqrt[c
+ d*x]) + (7*d)/(12*(b*c - a*d)^2*(a + b*x)^2*Sqrt[c + d*x]) - (35*d^2)/(24*(b*c
- a*d)^3*(a + b*x)*Sqrt[c + d*x]) + (35*Sqrt[b]*d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d
*x])/Sqrt[b*c - a*d]])/(8*(b*c - a*d)^(9/2))

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Rubi [A]  time = 0.189536, antiderivative size = 173, 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 \[ -\frac{35 d^3}{8 \sqrt{c+d x} (b c-a d)^4}+\frac{35 \sqrt{b} d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 (b c-a d)^{9/2}}-\frac{35 d^2}{24 (a+b x) \sqrt{c+d x} (b c-a d)^3}+\frac{7 d}{12 (a+b x)^2 \sqrt{c+d x} (b c-a d)^2}-\frac{1}{3 (a+b x)^3 \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^4*(c + d*x)^(3/2)),x]

[Out]

(-35*d^3)/(8*(b*c - a*d)^4*Sqrt[c + d*x]) - 1/(3*(b*c - a*d)*(a + b*x)^3*Sqrt[c
+ d*x]) + (7*d)/(12*(b*c - a*d)^2*(a + b*x)^2*Sqrt[c + d*x]) - (35*d^2)/(24*(b*c
- a*d)^3*(a + b*x)*Sqrt[c + d*x]) + (35*Sqrt[b]*d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d
*x])/Sqrt[b*c - a*d]])/(8*(b*c - a*d)^(9/2))

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Rubi in Sympy [A]  time = 42.1063, size = 153, normalized size = 0.88 \[ - \frac{35 \sqrt{b} d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{8 \left (a d - b c\right )^{\frac{9}{2}}} - \frac{35 d^{3}}{8 \sqrt{c + d x} \left (a d - b c\right )^{4}} + \frac{35 d^{2}}{24 \left (a + b x\right ) \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{7 d}{12 \left (a + b x\right )^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{1}{3 \left (a + b x\right )^{3} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**4/(d*x+c)**(3/2),x)

[Out]

-35*sqrt(b)*d**3*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(8*(a*d - b*c)**(9/
2)) - 35*d**3/(8*sqrt(c + d*x)*(a*d - b*c)**4) + 35*d**2/(24*(a + b*x)*sqrt(c +
d*x)*(a*d - b*c)**3) + 7*d/(12*(a + b*x)**2*sqrt(c + d*x)*(a*d - b*c)**2) + 1/(3
*(a + b*x)**3*sqrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.488105, size = 141, normalized size = 0.82 \[ \frac{35 \sqrt{b} d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left (-\frac{22 b d (b c-a d)}{(a+b x)^2}+\frac{8 b (b c-a d)^2}{(a+b x)^3}+\frac{57 b d^2}{a+b x}+\frac{48 d^3}{c+d x}\right )}{24 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^4*(c + d*x)^(3/2)),x]

[Out]

-(Sqrt[c + d*x]*((8*b*(b*c - a*d)^2)/(a + b*x)^3 - (22*b*d*(b*c - a*d))/(a + b*x
)^2 + (57*b*d^2)/(a + b*x) + (48*d^3)/(c + d*x)))/(24*(b*c - a*d)^4) + (35*Sqrt[
b]*d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*(b*c - a*d)^(9/2))

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Maple [B]  time = 0.029, size = 292, normalized size = 1.7 \[ -2\,{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{4}\sqrt{dx+c}}}-{\frac{19\,{d}^{3}{b}^{3}}{8\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{17\,{d}^{4}{b}^{2}a}{3\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{17\,{d}^{3}{b}^{3}c}{3\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{29\,{d}^{5}b{a}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}+{\frac{29\,{d}^{4}{b}^{2}ac}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}-{\frac{29\,{d}^{3}{b}^{3}{c}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}-{\frac{35\,{d}^{3}b}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^4/(d*x+c)^(3/2),x)

[Out]

-2*d^3/(a*d-b*c)^4/(d*x+c)^(1/2)-19/8*d^3/(a*d-b*c)^4*b^3/(b*d*x+a*d)^3*(d*x+c)^
(5/2)-17/3*d^4/(a*d-b*c)^4*b^2/(b*d*x+a*d)^3*(d*x+c)^(3/2)*a+17/3*d^3/(a*d-b*c)^
4*b^3/(b*d*x+a*d)^3*(d*x+c)^(3/2)*c-29/8*d^5/(a*d-b*c)^4*b/(b*d*x+a*d)^3*(d*x+c)
^(1/2)*a^2+29/4*d^4/(a*d-b*c)^4*b^2/(b*d*x+a*d)^3*(d*x+c)^(1/2)*a*c-29/8*d^3/(a*
d-b*c)^4*b^3/(b*d*x+a*d)^3*(d*x+c)^(1/2)*c^2-35/8*d^3/(a*d-b*c)^4*b/((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(1/((b*x + a)^4*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238393, 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)^4*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

[-1/48*(210*b^3*d^3*x^3 + 16*b^3*c^3 - 76*a*b^2*c^2*d + 174*a^2*b*c*d^2 + 96*a^3
*d^3 + 70*(b^3*c*d^2 + 8*a*b^2*d^3)*x^2 - 105*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3
*a^2*b*d^3*x + a^3*d^3)*sqrt(d*x + c)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a
*d + 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) - 14*(2*b^3*c^2
*d - 14*a*b^2*c*d^2 - 33*a^2*b*d^3)*x)/((a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b
^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4 + (b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*
d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^
3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*
c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*x)*sqrt(d*x + c)), -1/2
4*(105*b^3*d^3*x^3 + 8*b^3*c^3 - 38*a*b^2*c^2*d + 87*a^2*b*c*d^2 + 48*a^3*d^3 +
35*(b^3*c*d^2 + 8*a*b^2*d^3)*x^2 - 105*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*
d^3*x + a^3*d^3)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/
(b*c - a*d))/(sqrt(d*x + c)*b)) - 7*(2*b^3*c^2*d - 14*a*b^2*c*d^2 - 33*a^2*b*d^3
)*x)/((a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d
^4 + (b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^
4)*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 +
a^5*b^2*d^4)*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*
b^2*c*d^3 + a^6*b*d^4)*x)*sqrt(d*x + c))]

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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)**4/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223197, size = 440, normalized size = 2.54 \[ -\frac{35 \, b d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \, d^{3}}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{d x + c}} - \frac{57 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} d^{3} - 136 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d^{3} + 87 \, \sqrt{d x + c} b^{3} c^{2} d^{3} + 136 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{4} - 174 \, \sqrt{d x + c} a b^{2} c d^{4} + 87 \, \sqrt{d x + c} a^{2} b d^{5}}{24 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^4*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

-35/8*b*d^3*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)) - 2*d^3/
((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(d*
x + c)) - 1/24*(57*(d*x + c)^(5/2)*b^3*d^3 - 136*(d*x + c)^(3/2)*b^3*c*d^3 + 87*
sqrt(d*x + c)*b^3*c^2*d^3 + 136*(d*x + c)^(3/2)*a*b^2*d^4 - 174*sqrt(d*x + c)*a*
b^2*c*d^4 + 87*sqrt(d*x + c)*a^2*b*d^5)/((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)^3)