3.1432 \(\int \frac{1}{(a+b x)^3 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=140 \[ \frac{15 d^2}{4 \sqrt{c+d x} (b c-a d)^3}-\frac{15 \sqrt{b} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{7/2}}+\frac{5 d}{4 (a+b x) \sqrt{c+d x} (b c-a d)^2}-\frac{1}{2 (a+b x)^2 \sqrt{c+d x} (b c-a d)} \]

[Out]

(15*d^2)/(4*(b*c - a*d)^3*Sqrt[c + d*x]) - 1/(2*(b*c - a*d)*(a + b*x)^2*Sqrt[c +
d*x]) + (5*d)/(4*(b*c - a*d)^2*(a + b*x)*Sqrt[c + d*x]) - (15*Sqrt[b]*d^2*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.148287, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{15 d^2}{4 \sqrt{c+d x} (b c-a d)^3}-\frac{15 \sqrt{b} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{7/2}}+\frac{5 d}{4 (a+b x) \sqrt{c+d x} (b c-a d)^2}-\frac{1}{2 (a+b x)^2 \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(15*d^2)/(4*(b*c - a*d)^3*Sqrt[c + d*x]) - 1/(2*(b*c - a*d)*(a + b*x)^2*Sqrt[c +
d*x]) + (5*d)/(4*(b*c - a*d)^2*(a + b*x)*Sqrt[c + d*x]) - (15*Sqrt[b]*d^2*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 30.1572, size = 122, normalized size = 0.87 \[ - \frac{15 \sqrt{b} d^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 \left (a d - b c\right )^{\frac{7}{2}}} - \frac{15 d^{2}}{4 \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{5 d}{4 \left (a + b x\right ) \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(b)*d**2*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(4*(a*d - b*c)**(7/
2)) - 15*d**2/(4*sqrt(c + d*x)*(a*d - b*c)**3) + 5*d/(4*(a + b*x)*sqrt(c + d*x)*
(a*d - b*c)**2) + 1/(2*(a + b*x)**2*sqrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.323125, size = 126, normalized size = 0.9 \[ \frac{1}{4} \left (\frac{8 a^2 d^2+a b d (9 c+25 d x)+b^2 \left (-2 c^2+5 c d x+15 d^2 x^2\right )}{(a+b x)^2 \sqrt{c+d x} (b c-a d)^3}-\frac{15 \sqrt{b} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((8*a^2*d^2 + a*b*d*(9*c + 25*d*x) + b^2*(-2*c^2 + 5*c*d*x + 15*d^2*x^2))/((b*c
- a*d)^3*(a + b*x)^2*Sqrt[c + d*x]) - (15*Sqrt[b]*d^2*ArcTanh[(Sqrt[b]*Sqrt[c +
d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(7/2))/4

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Maple [A]  time = 0.025, size = 179, normalized size = 1.3 \[ -2\,{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}-{\frac{7\,{d}^{2}{b}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) ^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{d}^{3}ba}{4\, \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{9\,{d}^{2}{b}^{2}c}{4\, \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{15\,{d}^{2}b}{4\, \left ( ad-bc \right ) ^{3}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d^2/(a*d-b*c)^3/(d*x+c)^(1/2)-7/4*d^2/(a*d-b*c)^3*b^2/(b*d*x+a*d)^2*(d*x+c)^(
3/2)-9/4*d^3/(a*d-b*c)^3*b/(b*d*x+a*d)^2*(d*x+c)^(1/2)*a+9/4*d^2/(a*d-b*c)^3*b^2
/(b*d*x+a*d)^2*(d*x+c)^(1/2)*c-15/4*d^2/(a*d-b*c)^3*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)^3*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22767, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} d^{2} x^{2} - 4 \, b^{2} c^{2} + 18 \, a b c d + 16 \, a^{2} d^{2} - 15 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) + 10 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{8 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} +{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )} \sqrt{d x + c}}, \frac{15 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} + 9 \, a b c d + 8 \, a^{2} d^{2} - 15 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x + c} b}\right ) + 5 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{4 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} +{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )} \sqrt{d x + c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(30*b^2*d^2*x^2 - 4*b^2*c^2 + 18*a*b*c*d + 16*a^2*d^2 - 15*(b^2*d^2*x^2 + 2
*a*b*d^2*x + a^2*d^2)*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)) + 10*(b^2*c*d + 5
*a*b*d^2)*x)/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3 + (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)*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^
3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3)*x)*sqrt(d*x + c)), 1/4*(15*b^2*d^2*x^2 -
2*b^2*c^2 + 9*a*b*c*d + 8*a^2*d^2 - 15*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqr
t(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)) + 5*(b^2*c*d + 5*a*b*d^2)*x)/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4
*b*c*d^2 - a^5*d^3 + (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)*x
^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3)*x)*sqrt(d*x +
c))]

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Sympy [F(-2)]  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)**3/(d*x+c)**(3/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.220921, size = 316, normalized size = 2.26 \[ \frac{15 \, b d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \, d^{2}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{d x + c}} + \frac{7 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{2} - 9 \, \sqrt{d x + c} b^{2} c d^{2} + 9 \, \sqrt{d x + c} a b d^{3}}{4 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/4*b*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c^3 - 3*a*b^2*c^2*
d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b^2*c + a*b*d)) + 2*d^2/((b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(d*x + c)) + 1/4*(7*(d*x + c)^(3/2)*b^2*d^2
- 9*sqrt(d*x + c)*b^2*c*d^2 + 9*sqrt(d*x + c)*a*b*d^3)/((b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*((d*x + c)*b - b*c + a*d)^2)