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

Optimal. Leaf size=240 \[ -\frac{d \sqrt{a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{24 a c^4 \sqrt{c+d x}}+\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{9/2}}-\frac{\sqrt{a+b x} (3 b c-35 a d) (b c-a d)}{24 a c^3 x \sqrt{c+d x}}-\frac{7 \sqrt{a+b x} (b c-a d)}{12 c^2 x^2 \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}} \]

[Out]

-(d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a + b*x])/(24*a*c^4*Sqrt[c + d*
x]) - (a*Sqrt[a + b*x])/(3*c*x^3*Sqrt[c + d*x]) - (7*(b*c - a*d)*Sqrt[a + b*x])/
(12*c^2*x^2*Sqrt[c + d*x]) - ((3*b*c - 35*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(24*a*
c^3*x*Sqrt[c + d*x]) + ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[
(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*c^(9/2))

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Rubi [A]  time = 0.785005, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{d \sqrt{a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{24 a c^4 \sqrt{c+d x}}+\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{9/2}}-\frac{\sqrt{a+b x} (3 b c-35 a d) (b c-a d)}{24 a c^3 x \sqrt{c+d x}}-\frac{7 \sqrt{a+b x} (b c-a d)}{12 c^2 x^2 \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a + b*x])/(24*a*c^4*Sqrt[c + d*
x]) - (a*Sqrt[a + b*x])/(3*c*x^3*Sqrt[c + d*x]) - (7*(b*c - a*d)*Sqrt[a + b*x])/
(12*c^2*x^2*Sqrt[c + d*x]) - ((3*b*c - 35*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(24*a*
c^3*x*Sqrt[c + d*x]) + ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[
(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*c^(9/2))

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Rubi in Sympy [A]  time = 118.695, size = 223, normalized size = 0.93 \[ - \frac{a \sqrt{a + b x}}{3 c x^{3} \sqrt{c + d x}} + \frac{7 \sqrt{a + b x} \left (a d - b c\right )}{12 c^{2} x^{2} \sqrt{c + d x}} - \frac{\sqrt{a + b x} \left (a d - b c\right ) \left (35 a d - 3 b c\right )}{24 a c^{3} x \sqrt{c + d x}} - \frac{d \sqrt{a + b x} \left (105 a^{2} d^{2} - 100 a b c d + 3 b^{2} c^{2}\right )}{24 a c^{4} \sqrt{c + d x}} + \frac{\left (a d - b c\right ) \left (35 a^{2} d^{2} - 10 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{3}{2}} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(a + b*x)/(3*c*x**3*sqrt(c + d*x)) + 7*sqrt(a + b*x)*(a*d - b*c)/(12*c**2
*x**2*sqrt(c + d*x)) - sqrt(a + b*x)*(a*d - b*c)*(35*a*d - 3*b*c)/(24*a*c**3*x*s
qrt(c + d*x)) - d*sqrt(a + b*x)*(105*a**2*d**2 - 100*a*b*c*d + 3*b**2*c**2)/(24*
a*c**4*sqrt(c + d*x)) + (a*d - b*c)*(35*a**2*d**2 - 10*a*b*c*d - b**2*c**2)*atan
h(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a**(3/2)*c**(9/2))

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Mathematica [A]  time = 0.315555, size = 234, normalized size = 0.98 \[ \frac{-3 \log (x) (b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )+3 (b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-\frac{2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \left (a^2 \left (8 c^3-14 c^2 d x+35 c d^2 x^2+105 d^3 x^3\right )+2 a b c x \left (7 c^2-19 c d x-50 d^2 x^2\right )+3 b^2 c^2 x^2 (c+d x)\right )}{x^3 \sqrt{c+d x}}}{48 a^{3/2} c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*(3*b^2*c^2*x^2*(c + d*x) + 2*a*b*c*x*(7*c^2 -
19*c*d*x - 50*d^2*x^2) + a^2*(8*c^3 - 14*c^2*d*x + 35*c*d^2*x^2 + 105*d^3*x^3))
)/(x^3*Sqrt[c + d*x]) - 3*(b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Log[x]
+ 3*(b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*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]])/(48*a^(3/2)*c^(9/2))

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Maple [B]  time = 0.046, size = 707, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(105*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*d^4-135*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*d^3+27*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^2*d^2+3*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^3*d+105*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*d^3-135*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^2*d^2+27*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^3*d+3*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^4-210*x^3*a^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)+200*x^3*a*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x^3*b^2*c^2*d*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-70*x^2*a^2*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+76*x^2*a*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x^2*b^2*c^3*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+28*x*a^2*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)-28*x*a*b*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-16*a^2*c^3*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2))/c^4/a/((b*x+a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2)/(d*x+c)^(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((b*x + a)^(3/2)/((d*x + c)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.710802, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (8 \, a^{2} c^{3} +{\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{2} + 14 \,{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (b^{3} c^{3} d + 9 \, a b^{2} c^{2} d^{2} - 45 \, a^{2} b c d^{3} + 35 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} + 9 \, a b^{2} c^{3} d - 45 \, a^{2} b c^{2} d^{2} + 35 \, a^{3} c d^{3}\right )} x^{3}\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{96 \,{\left (a c^{4} d x^{4} + a c^{5} x^{3}\right )} \sqrt{a c}}, -\frac{2 \,{\left (8 \, a^{2} c^{3} +{\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{2} + 14 \,{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (b^{3} c^{3} d + 9 \, a b^{2} c^{2} d^{2} - 45 \, a^{2} b c d^{3} + 35 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} + 9 \, a b^{2} c^{3} d - 45 \, a^{2} b c^{2} d^{2} + 35 \, a^{3} c d^{3}\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{48 \,{\left (a c^{4} d x^{4} + a c^{5} x^{3}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/96*(4*(8*a^2*c^3 + (3*b^2*c^2*d - 100*a*b*c*d^2 + 105*a^2*d^3)*x^3 + (3*b^2*
c^3 - 38*a*b*c^2*d + 35*a^2*c*d^2)*x^2 + 14*(a*b*c^3 - a^2*c^2*d)*x)*sqrt(a*c)*s
qrt(b*x + a)*sqrt(d*x + c) - 3*((b^3*c^3*d + 9*a*b^2*c^2*d^2 - 45*a^2*b*c*d^3 +
35*a^3*d^4)*x^4 + (b^3*c^4 + 9*a*b^2*c^3*d - 45*a^2*b*c^2*d^2 + 35*a^3*c*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))/((a*c^4*d*x^4 + a*c^5*x^3)*sqrt(a*c)), -1/48*(2*(8*a^2*c^3 + (3*b^2*c^2
*d - 100*a*b*c*d^2 + 105*a^2*d^3)*x^3 + (3*b^2*c^3 - 38*a*b*c^2*d + 35*a^2*c*d^2
)*x^2 + 14*(a*b*c^3 - a^2*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(
(b^3*c^3*d + 9*a*b^2*c^2*d^2 - 45*a^2*b*c*d^3 + 35*a^3*d^4)*x^4 + (b^3*c^4 + 9*a
*b^2*c^3*d - 45*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*x^3)*arctan(1/2*(2*a*c + (b*c + a*
d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((a*c^4*d*x^4 + a*c^5*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)**(3/2)/x**4/(d*x+c)**(3/2),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)^(3/2)/((d*x + c)^(3/2)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError