3.1442 \(\int \frac{1}{(a+b x)^3 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{35 b^{3/2} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{9/2}}+\frac{35 b d^2}{4 \sqrt{c+d x} (b c-a d)^4}+\frac{35 d^2}{12 (c+d x)^{3/2} (b c-a d)^3}+\frac{7 d}{4 (a+b x) (c+d x)^{3/2} (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.194914, antiderivative size = 167, 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 \[ -\frac{35 b^{3/2} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{9/2}}+\frac{35 b d^2}{4 \sqrt{c+d x} (b c-a d)^4}+\frac{35 d^2}{12 (c+d x)^{3/2} (b c-a d)^3}+\frac{7 d}{4 (a+b x) (c+d x)^{3/2} (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^3*(c + d*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.5389, size = 148, normalized size = 0.89 \[ \frac{35 b^{\frac{3}{2}} d^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 \left (a d - b c\right )^{\frac{9}{2}}} + \frac{35 b d^{2}}{4 \sqrt{c + d x} \left (a d - b c\right )^{4}} - \frac{35 d^{2}}{12 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{7 d}{4 \left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**3/(d*x+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.474677, size = 143, normalized size = 0.86 \[ \frac{\sqrt{c+d x} \left (-\frac{6 b^2 (b c-a d)}{(a+b x)^2}+\frac{33 b^2 d}{a+b x}+\frac{8 d^2 (b c-a d)}{(c+d x)^2}+\frac{72 b d^2}{c+d x}\right )}{12 (b c-a d)^4}-\frac{35 b^{3/2} d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 (b c-a d)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^3*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[c + d*x]*((-6*b^2*(b*c - a*d))/(a + b*x)^2 + (33*b^2*d)/(a + b*x) + (8*d^2
*(b*c - a*d))/(c + d*x)^2 + (72*b*d^2)/(c + d*x)))/(12*(b*c - a*d)^4) - (35*b^(3
/2)*d^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*(b*c - a*d)^(9/2))

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Maple [A]  time = 0.027, size = 206, normalized size = 1.2 \[ -{\frac{2\,{d}^{2}}{3\, \left ( ad-bc \right ) ^{3}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{{d}^{2}b}{ \left ( ad-bc \right ) ^{4}\sqrt{dx+c}}}+{\frac{11\,{d}^{2}{b}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{d}^{3}{b}^{2}a}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{13\,{d}^{2}{b}^{3}c}{4\, \left ( ad-bc \right ) ^{4} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{35\,{d}^{2}{b}^{2}}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^3/(d*x+c)^(5/2),x)

[Out]

-2/3*d^2/(a*d-b*c)^3/(d*x+c)^(3/2)+6*d^2/(a*d-b*c)^4*b/(d*x+c)^(1/2)+11/4*d^2/(a
*d-b*c)^4*b^3/(b*d*x+a*d)^2*(d*x+c)^(3/2)+13/4*d^3/(a*d-b*c)^4*b^2/(b*d*x+a*d)^2
*(d*x+c)^(1/2)*a-13/4*d^2/(a*d-b*c)^4*b^3/(b*d*x+a*d)^2*(d*x+c)^(1/2)*c+35/4*d^2
/(a*d-b*c)^4*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(1/((b*x + a)^3*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236781, 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)^3*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/24*(210*b^3*d^3*x^3 - 12*b^3*c^3 + 78*a*b^2*c^2*d + 160*a^2*b*c*d^2 - 16*a^3*
d^3 + 70*(4*b^3*c*d^2 + 5*a*b^2*d^3)*x^2 + 105*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3
*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*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*(3*b^3*c^2*d + 34*a*b^2*c*d^2 + 8*a^2*b*d^3)*x)/((a^2*b
^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2 - 4*a^5*b*c^2*d^3 + a^6*c*d^4 + (b^
6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^4*b^2*d^5)*x
^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3*d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^4*b^
2*c*d^4 + 2*a^5*b*d^5)*x^2 + (2*a*b^5*c^5 - 7*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2
- 2*a^4*b^2*c^2*d^3 - 2*a^5*b*c*d^4 + a^6*d^5)*x)*sqrt(d*x + c)), 1/12*(105*b^3*
d^3*x^3 - 6*b^3*c^3 + 39*a*b^2*c^2*d + 80*a^2*b*c*d^2 - 8*a^3*d^3 + 35*(4*b^3*c*
d^2 + 5*a*b^2*d^3)*x^2 - 105*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d
^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))*arct
an(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x + c)*b)) + 7*(3*b^3*c^2*d + 34*a*
b^2*c*d^2 + 8*a^2*b*d^3)*x)/((a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2
- 4*a^5*b*c^2*d^3 + a^6*c*d^4 + (b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3
- 4*a^3*b^3*c*d^4 + a^4*b^2*d^5)*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3
*d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^4*b^2*c*d^4 + 2*a^5*b*d^5)*x^2 + (2*a*b^5*c^5 - 7
*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^3 - 2*a^5*b*c*d^4 + a^6*d^5
)*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)**(5/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.223968, size = 402, normalized size = 2.41 \[ \frac{35 \, b^{2} d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (9 \,{\left (d x + c\right )} b d^{2} + b c d^{2} - a d^{3}\right )}}{3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} + \frac{11 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} d^{2} - 13 \, \sqrt{d x + c} b^{3} c d^{2} + 13 \, \sqrt{d x + c} a b^{2} d^{3}}{4 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

35/4*b^2*d^2*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)) + 2/3*(
9*(d*x + c)*b*d^2 + b*c*d^2 - a*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)*(d*x + c)^(3/2)) + 1/4*(11*(d*x + c)^(3/2)*b^3*d^2
- 13*sqrt(d*x + c)*b^3*c*d^2 + 13*sqrt(d*x + c)*a*b^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)*((d*x + c)*b - b*c + a*d)^2
)