3.1443 \(\int \frac{1}{(a+b x)^4 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=200 \[ \frac{105 b^{3/2} d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 (b c-a d)^{11/2}}-\frac{105 b d^3}{8 \sqrt{c+d x} (b c-a d)^5}-\frac{35 d^3}{8 (c+d x)^{3/2} (b c-a d)^4}-\frac{21 d^2}{8 (a+b x) (c+d x)^{3/2} (b c-a d)^3}+\frac{3 d}{4 (a+b x)^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (c+d x)^{3/2} (b c-a d)} \]

[Out]

(-35*d^3)/(8*(b*c - a*d)^4*(c + d*x)^(3/2)) - 1/(3*(b*c - a*d)*(a + b*x)^3*(c +
d*x)^(3/2)) + (3*d)/(4*(b*c - a*d)^2*(a + b*x)^2*(c + d*x)^(3/2)) - (21*d^2)/(8*
(b*c - a*d)^3*(a + b*x)*(c + d*x)^(3/2)) - (105*b*d^3)/(8*(b*c - a*d)^5*Sqrt[c +
d*x]) + (105*b^(3/2)*d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*(
b*c - a*d)^(11/2))

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Rubi [A]  time = 0.376661, antiderivative size = 200, 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 \[ \frac{105 b^{3/2} d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 (b c-a d)^{11/2}}-\frac{105 b d^3}{8 \sqrt{c+d x} (b c-a d)^5}-\frac{35 d^3}{8 (c+d x)^{3/2} (b c-a d)^4}-\frac{21 d^2}{8 (a+b x) (c+d x)^{3/2} (b c-a d)^3}+\frac{3 d}{4 (a+b x)^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^4*(c + d*x)^(5/2)),x]

[Out]

(-35*d^3)/(8*(b*c - a*d)^4*(c + d*x)^(3/2)) - 1/(3*(b*c - a*d)*(a + b*x)^3*(c +
d*x)^(3/2)) + (3*d)/(4*(b*c - a*d)^2*(a + b*x)^2*(c + d*x)^(3/2)) - (21*d^2)/(8*
(b*c - a*d)^3*(a + b*x)*(c + d*x)^(3/2)) - (105*b*d^3)/(8*(b*c - a*d)^5*Sqrt[c +
d*x]) + (105*b^(3/2)*d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*(
b*c - a*d)^(11/2))

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Rubi in Sympy [A]  time = 51.9346, size = 178, normalized size = 0.89 \[ \frac{105 b^{\frac{3}{2}} d^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{8 \left (a d - b c\right )^{\frac{11}{2}}} + \frac{105 b d^{3}}{8 \sqrt{c + d x} \left (a d - b c\right )^{5}} - \frac{35 d^{3}}{8 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{21 d^{2}}{8 \left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{3 d}{4 \left (a + b x\right )^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{1}{3 \left (a + b x\right )^{3} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**4/(d*x+c)**(5/2),x)

[Out]

105*b**(3/2)*d**3*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(8*(a*d - b*c)**(1
1/2)) + 105*b*d**3/(8*sqrt(c + d*x)*(a*d - b*c)**5) - 35*d**3/(8*(c + d*x)**(3/2
)*(a*d - b*c)**4) + 21*d**2/(8*(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**3) + 3*d/
(4*(a + b*x)**2*(c + d*x)**(3/2)*(a*d - b*c)**2) + 1/(3*(a + b*x)**3*(c + d*x)**
(3/2)*(a*d - b*c))

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Mathematica [A]  time = 0.821077, size = 167, normalized size = 0.84 \[ \frac{1}{24} \left (\frac{315 b^{3/2} d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{11/2}}+\frac{\sqrt{c+d x} \left (\frac{34 b^2 d (b c-a d)}{(a+b x)^2}-\frac{8 b^2 (b c-a d)^2}{(a+b x)^3}-\frac{123 b^2 d^2}{a+b x}+\frac{16 d^3 (a d-b c)}{(c+d x)^2}-\frac{192 b d^3}{c+d x}\right )}{(b c-a d)^5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^4*(c + d*x)^(5/2)),x]

[Out]

((Sqrt[c + d*x]*((-8*b^2*(b*c - a*d)^2)/(a + b*x)^3 + (34*b^2*d*(b*c - a*d))/(a
+ b*x)^2 - (123*b^2*d^2)/(a + b*x) + (16*d^3*(-(b*c) + a*d))/(c + d*x)^2 - (192*
b*d^3)/(c + d*x)))/(b*c - a*d)^5 + (315*b^(3/2)*d^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*
x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(11/2))/24

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Maple [A]  time = 0.032, size = 319, normalized size = 1.6 \[ -{\frac{2\,{d}^{3}}{3\, \left ( ad-bc \right ) ^{4}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+8\,{\frac{{d}^{3}b}{ \left ( ad-bc \right ) ^{5}\sqrt{dx+c}}}+{\frac{41\,{d}^{3}{b}^{4}}{8\, \left ( ad-bc \right ) ^{5} \left ( bdx+ad \right ) ^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{d}^{4}{b}^{3}a}{3\, \left ( ad-bc \right ) ^{5} \left ( bdx+ad \right ) ^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{d}^{3}{b}^{4}c}{3\, \left ( ad-bc \right ) ^{5} \left ( bdx+ad \right ) ^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{d}^{5}{b}^{2}{a}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}-{\frac{55\,{d}^{4}{b}^{3}ac}{4\, \left ( ad-bc \right ) ^{5} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}+{\frac{55\,{d}^{3}{b}^{4}{c}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}+{\frac{105\,{d}^{3}{b}^{2}}{8\, \left ( ad-bc \right ) ^{5}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^4/(d*x+c)^(5/2),x)

[Out]

-2/3*d^3/(a*d-b*c)^4/(d*x+c)^(3/2)+8*d^3/(a*d-b*c)^5*b/(d*x+c)^(1/2)+41/8*d^3/(a
*d-b*c)^5*b^4/(b*d*x+a*d)^3*(d*x+c)^(5/2)+35/3*d^4/(a*d-b*c)^5*b^3/(b*d*x+a*d)^3
*(d*x+c)^(3/2)*a-35/3*d^3/(a*d-b*c)^5*b^4/(b*d*x+a*d)^3*(d*x+c)^(3/2)*c+55/8*d^5
/(a*d-b*c)^5*b^2/(b*d*x+a*d)^3*(d*x+c)^(1/2)*a^2-55/4*d^4/(a*d-b*c)^5*b^3/(b*d*x
+a*d)^3*(d*x+c)^(1/2)*a*c+55/8*d^3/(a*d-b*c)^5*b^4/(b*d*x+a*d)^3*(d*x+c)^(1/2)*c
^2+105/8*d^3/(a*d-b*c)^5*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)^4*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246455, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^4*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(630*b^4*d^4*x^4 + 16*b^4*c^4 - 100*a*b^3*c^3*d + 330*a^2*b^2*c^2*d^2 + 4
16*a^3*b*c*d^3 - 32*a^4*d^4 + 840*(b^4*c*d^3 + 2*a*b^3*d^4)*x^3 + 126*(b^4*c^2*d
^2 + 18*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 + 315*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^
4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^
3 + a^3*b*d^4)*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)) - 36*(b^4*c^3*d - 10*
a*b^3*c^2*d^2 - 53*a^2*b^2*c*d^3 - 8*a^3*b*d^4)*x)/((a^3*b^5*c^6 - 5*a^4*b^4*c^5
*d + 10*a^5*b^3*c^4*d^2 - 10*a^6*b^2*c^3*d^3 + 5*a^7*b*c^2*d^4 - a^8*c*d^5 + (b^
8*c^5*d - 5*a*b^7*c^4*d^2 + 10*a^2*b^6*c^3*d^3 - 10*a^3*b^5*c^2*d^4 + 5*a^4*b^4*
c*d^5 - a^5*b^3*d^6)*x^4 + (b^8*c^6 - 2*a*b^7*c^5*d - 5*a^2*b^6*c^4*d^2 + 20*a^3
*b^5*c^3*d^3 - 25*a^4*b^4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2*d^6)*x^3 + 3*(a
*b^7*c^6 - 4*a^2*b^6*c^5*d + 5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2*d^4 + 4*a^6*b^2*c
*d^5 - a^7*b*d^6)*x^2 + (3*a^2*b^6*c^6 - 14*a^3*b^5*c^5*d + 25*a^4*b^4*c^4*d^2 -
20*a^5*b^3*c^3*d^3 + 5*a^6*b^2*c^2*d^4 + 2*a^7*b*c*d^5 - a^8*d^6)*x)*sqrt(d*x +
c)), -1/24*(315*b^4*d^4*x^4 + 8*b^4*c^4 - 50*a*b^3*c^3*d + 165*a^2*b^2*c^2*d^2
+ 208*a^3*b*c*d^3 - 16*a^4*d^4 + 420*(b^4*c*d^3 + 2*a*b^3*d^4)*x^3 + 63*(b^4*c^2
*d^2 + 18*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 315*(b^4*d^4*x^4 + a^3*b*c*d^3 + (
b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*
d^3 + a^3*b*d^4)*x)*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)) - 18*(b^4*c^3*d - 10*a*b^3*c^2*d^2 - 53*a^2*b
^2*c*d^3 - 8*a^3*b*d^4)*x)/((a^3*b^5*c^6 - 5*a^4*b^4*c^5*d + 10*a^5*b^3*c^4*d^2
- 10*a^6*b^2*c^3*d^3 + 5*a^7*b*c^2*d^4 - a^8*c*d^5 + (b^8*c^5*d - 5*a*b^7*c^4*d^
2 + 10*a^2*b^6*c^3*d^3 - 10*a^3*b^5*c^2*d^4 + 5*a^4*b^4*c*d^5 - a^5*b^3*d^6)*x^4
+ (b^8*c^6 - 2*a*b^7*c^5*d - 5*a^2*b^6*c^4*d^2 + 20*a^3*b^5*c^3*d^3 - 25*a^4*b^
4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2*d^6)*x^3 + 3*(a*b^7*c^6 - 4*a^2*b^6*c^5
*d + 5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2*d^4 + 4*a^6*b^2*c*d^5 - a^7*b*d^6)*x^2 +
(3*a^2*b^6*c^6 - 14*a^3*b^5*c^5*d + 25*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 5*
a^6*b^2*c^2*d^4 + 2*a^7*b*c*d^5 - a^8*d^6)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227401, size = 583, normalized size = 2.92 \[ -\frac{105 \, b^{2} d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt{-b^{2} c + a b d}} - \frac{315 \,{\left (d x + c\right )}^{4} b^{4} d^{3} - 840 \,{\left (d x + c\right )}^{3} b^{4} c d^{3} + 693 \,{\left (d x + c\right )}^{2} b^{4} c^{2} d^{3} - 144 \,{\left (d x + c\right )} b^{4} c^{3} d^{3} - 16 \, b^{4} c^{4} d^{3} + 840 \,{\left (d x + c\right )}^{3} a b^{3} d^{4} - 1386 \,{\left (d x + c\right )}^{2} a b^{3} c d^{4} + 432 \,{\left (d x + c\right )} a b^{3} c^{2} d^{4} + 64 \, a b^{3} c^{3} d^{4} + 693 \,{\left (d x + c\right )}^{2} a^{2} b^{2} d^{5} - 432 \,{\left (d x + c\right )} a^{2} b^{2} c d^{5} - 96 \, a^{2} b^{2} c^{2} d^{5} + 144 \,{\left (d x + c\right )} a^{3} b d^{6} + 64 \, a^{3} b c d^{6} - 16 \, a^{4} d^{7}}{24 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}{\left ({\left (d x + c\right )}^{\frac{3}{2}} b - \sqrt{d x + c} b c + \sqrt{d x + c} a d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^4*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

-105/8*b^2*d^3*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^5 - 5*a*b^4*
c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(
-b^2*c + a*b*d)) - 1/24*(315*(d*x + c)^4*b^4*d^3 - 840*(d*x + c)^3*b^4*c*d^3 + 6
93*(d*x + c)^2*b^4*c^2*d^3 - 144*(d*x + c)*b^4*c^3*d^3 - 16*b^4*c^4*d^3 + 840*(d
*x + c)^3*a*b^3*d^4 - 1386*(d*x + c)^2*a*b^3*c*d^4 + 432*(d*x + c)*a*b^3*c^2*d^4
+ 64*a*b^3*c^3*d^4 + 693*(d*x + c)^2*a^2*b^2*d^5 - 432*(d*x + c)*a^2*b^2*c*d^5
- 96*a^2*b^2*c^2*d^5 + 144*(d*x + c)*a^3*b*d^6 + 64*a^3*b*c*d^6 - 16*a^4*d^7)/((
b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^
4 - a^5*d^5)*((d*x + c)^(3/2)*b - sqrt(d*x + c)*b*c + sqrt(d*x + c)*a*d)^3)