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

Optimal. Leaf size=278 \[ -\frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{11/2}}-\frac{d \sqrt{a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{24 c^5 \sqrt{c+d x}}-\frac{7 d \sqrt{a+b x} (7 b c-15 a d) (b c-a d)}{24 c^4 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (11 b c-21 a d) (b c-a d)}{8 c^3 x (c+d x)^{3/2}}-\frac{3 a \sqrt{a+b x} (b c-a d)}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}} \]

[Out]

(-7*d*(7*b*c - 15*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(24*c^4*(c + d*x)^(3/2)) - (3*
a*(b*c - a*d)*Sqrt[a + b*x])/(4*c^2*x^2*(c + d*x)^(3/2)) - ((11*b*c - 21*a*d)*(b
*c - a*d)*Sqrt[a + b*x])/(8*c^3*x*(c + d*x)^(3/2)) - (a*(a + b*x)^(3/2))/(3*c*x^
3*(c + d*x)^(3/2)) - (d*(113*b^2*c^2 - 420*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])
/(24*c^5*Sqrt[c + d*x]) - (5*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(11/2))

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Rubi [A]  time = 1.14582, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{11/2}}-\frac{d \sqrt{a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{24 c^5 \sqrt{c+d x}}-\frac{7 d \sqrt{a+b x} (7 b c-15 a d) (b c-a d)}{24 c^4 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (11 b c-21 a d) (b c-a d)}{8 c^3 x (c+d x)^{3/2}}-\frac{3 a \sqrt{a+b x} (b c-a d)}{4 c^2 x^2 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*d*(7*b*c - 15*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(24*c^4*(c + d*x)^(3/2)) - (3*
a*(b*c - a*d)*Sqrt[a + b*x])/(4*c^2*x^2*(c + d*x)^(3/2)) - ((11*b*c - 21*a*d)*(b
*c - a*d)*Sqrt[a + b*x])/(8*c^3*x*(c + d*x)^(3/2)) - (a*(a + b*x)^(3/2))/(3*c*x^
3*(c + d*x)^(3/2)) - (d*(113*b^2*c^2 - 420*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])
/(24*c^5*Sqrt[c + d*x]) - (5*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(11/2))

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Rubi in Sympy [A]  time = 160.731, size = 265, normalized size = 0.95 \[ - \frac{a \left (a + b x\right )^{\frac{3}{2}}}{3 c x^{3} \left (c + d x\right )^{\frac{3}{2}}} + \frac{3 a \sqrt{a + b x} \left (a d - b c\right )}{4 c^{2} x^{2} \left (c + d x\right )^{\frac{3}{2}}} - \frac{\sqrt{a + b x} \left (a d - b c\right ) \left (21 a d - 11 b c\right )}{8 c^{3} x \left (c + d x\right )^{\frac{3}{2}}} - \frac{7 d \sqrt{a + b x} \left (a d - b c\right ) \left (15 a d - 7 b c\right )}{24 c^{4} \left (c + d x\right )^{\frac{3}{2}}} - \frac{d \sqrt{a + b x} \left (315 a^{2} d^{2} - 420 a b c d + 113 b^{2} c^{2}\right )}{24 c^{5} \sqrt{c + d x}} + \frac{5 \left (a d - b c\right ) \left (21 a^{2} d^{2} - 14 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 \sqrt{a} c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + b*x)**(3/2)/(3*c*x**3*(c + d*x)**(3/2)) + 3*a*sqrt(a + b*x)*(a*d - b*c)/
(4*c**2*x**2*(c + d*x)**(3/2)) - sqrt(a + b*x)*(a*d - b*c)*(21*a*d - 11*b*c)/(8*
c**3*x*(c + d*x)**(3/2)) - 7*d*sqrt(a + b*x)*(a*d - b*c)*(15*a*d - 7*b*c)/(24*c*
*4*(c + d*x)**(3/2)) - d*sqrt(a + b*x)*(315*a**2*d**2 - 420*a*b*c*d + 113*b**2*c
**2)/(24*c**5*sqrt(c + d*x)) + 5*(a*d - b*c)*(21*a**2*d**2 - 14*a*b*c*d + b**2*c
**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*sqrt(a)*c**(11/2))

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Mathematica [A]  time = 0.457434, size = 269, normalized size = 0.97 \[ \frac{\frac{15 \log (x) (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{\sqrt{a}}+\frac{15 (a d-b c) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}-\frac{2 \sqrt{c} \sqrt{a+b x} \left (a^2 \left (8 c^4-18 c^3 d x+63 c^2 d^2 x^2+420 c d^3 x^3+315 d^4 x^4\right )-2 a b c x \left (-13 c^3+48 c^2 d x+287 c d^2 x^2+210 d^3 x^3\right )+b^2 c^2 x^2 \left (33 c^2+162 c d x+113 d^2 x^2\right )\right )}{x^3 (c+d x)^{3/2}}}{48 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*Sqrt[a + b*x]*(b^2*c^2*x^2*(33*c^2 + 162*c*d*x + 113*d^2*x^2) - 2*a
*b*c*x*(-13*c^3 + 48*c^2*d*x + 287*c*d^2*x^2 + 210*d^3*x^3) + a^2*(8*c^4 - 18*c^
3*d*x + 63*c^2*d^2*x^2 + 420*c*d^3*x^3 + 315*d^4*x^4)))/(x^3*(c + d*x)^(3/2)) +
(15*(b*c - a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Log[x])/Sqrt[a] + (15*(-(b*c
) + a*d)*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[
a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[a])/(48*c^(11/2))

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Maple [B]  time = 0.05, size = 1009, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(315*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*
a*c)/x)*x^5*a^3*d^5-525*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*
a*c)/x)*x^5*a^2*b*c*d^4+225*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)+2*a*c)/x)*x^5*a*b^2*c^2*d^3-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+2*a*c)/x)*x^5*b^3*c^3*d^2+630*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+2*a*c)/x)*x^4*a^3*c*d^4-1050*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b*c^2*d^3+450*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a*b^2*c^3*d^2-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*b^3*c^4*d+315*ln((a*d*x+b*c*x+2*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^3*c^2*d^3-525*ln((a*d*x+b*c*x+2*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^2*b*c^3*d^2+225*ln((a*d*x+b*c*x+2
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a*b^2*c^4*d-15*ln((a*d*x+b*c*
x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*b^3*c^5-630*x^4*a^2*d^4*((
b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+840*x^4*a*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c
)^(1/2)-226*x^4*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-840*x^3*a^2*c*d^
3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+1148*x^3*a*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/
2)*(a*c)^(1/2)-324*x^3*b^2*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-126*x^2*a^2
*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+192*x^2*a*b*c^3*d*((b*x+a)*(d*x+c))
^(1/2)*(a*c)^(1/2)-66*x^2*b^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+36*x*a^2*c
^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-52*x*a*b*c^4*((b*x+a)*(d*x+c))^(1/2)*(a
*c)^(1/2)-16*a^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2))/c^5/((b*x+a)*(d*x+c))^
(1/2)/(a*c)^(1/2)/x^3/(d*x+c)^(3/2)

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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((b*x + a)^(5/2)/((d*x + c)^(5/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90873, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/96*(4*(8*a^2*c^4 + (113*b^2*c^2*d^2 - 420*a*b*c*d^3 + 315*a^2*d^4)*x^4 + 2*(
81*b^2*c^3*d - 287*a*b*c^2*d^2 + 210*a^2*c*d^3)*x^3 + 3*(11*b^2*c^4 - 32*a*b*c^3
*d + 21*a^2*c^2*d^2)*x^2 + 2*(13*a*b*c^4 - 9*a^2*c^3*d)*x)*sqrt(a*c)*sqrt(b*x +
a)*sqrt(d*x + c) + 15*((b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 35*a^2*b*c*d^4 - 21*a^3
*d^5)*x^5 + 2*(b^3*c^4*d - 15*a*b^2*c^3*d^2 + 35*a^2*b*c^2*d^3 - 21*a^3*c*d^4)*x
^4 + (b^3*c^5 - 15*a*b^2*c^4*d + 35*a^2*b*c^3*d^2 - 21*a^3*c^2*d^3)*x^3)*log((4*
(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (
b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2))/(
(c^5*d^2*x^5 + 2*c^6*d*x^4 + c^7*x^3)*sqrt(a*c)), -1/48*(2*(8*a^2*c^4 + (113*b^2
*c^2*d^2 - 420*a*b*c*d^3 + 315*a^2*d^4)*x^4 + 2*(81*b^2*c^3*d - 287*a*b*c^2*d^2
+ 210*a^2*c*d^3)*x^3 + 3*(11*b^2*c^4 - 32*a*b*c^3*d + 21*a^2*c^2*d^2)*x^2 + 2*(1
3*a*b*c^4 - 9*a^2*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((b^3*c^
3*d^2 - 15*a*b^2*c^2*d^3 + 35*a^2*b*c*d^4 - 21*a^3*d^5)*x^5 + 2*(b^3*c^4*d - 15*
a*b^2*c^3*d^2 + 35*a^2*b*c^2*d^3 - 21*a^3*c*d^4)*x^4 + (b^3*c^5 - 15*a*b^2*c^4*d
+ 35*a^2*b*c^3*d^2 - 21*a^3*c^2*d^3)*x^3)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sq
rt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((c^5*d^2*x^5 + 2*c^6*d*x^4 + c^7*x
^3)*sqrt(-a*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((b*x+a)**(5/2)/x**4/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError