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

Optimal. Leaf size=241 \[ -\frac{\sqrt{c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt{a+b x}}+\frac{7 \sqrt{c+d x} (b c-a d)}{12 a^2 x^2 \sqrt{a+b x}}+\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac{b \sqrt{c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.802934, antiderivative size = 241, 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 \[ -\frac{\sqrt{c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt{a+b x}}+\frac{7 \sqrt{c+d x} (b c-a d)}{12 a^2 x^2 \sqrt{a+b x}}+\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac{b \sqrt{c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 116.928, size = 223, normalized size = 0.93 \[ - \frac{c \sqrt{c + d x}}{3 a x^{3} \sqrt{a + b x}} - \frac{7 \sqrt{c + d x} \left (a d - b c\right )}{12 a^{2} x^{2} \sqrt{a + b x}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (3 a d - 35 b c\right )}{24 a^{3} c x \sqrt{a + b x}} - \frac{b \sqrt{c + d x} \left (3 a^{2} d^{2} - 100 a b c d + 105 b^{2} c^{2}\right )}{24 a^{4} c \sqrt{a + b x}} + \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{9}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.297871, size = 236, normalized size = 0.98 \[ \frac{-3 \log (x) (a d-b c) \left (a^2 d^2+10 a b c d-35 b^2 c^2\right )+3 (a d-b c) \left (a^2 d^2+10 a b c d-35 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-\frac{2 \sqrt{a} \sqrt{c} \sqrt{c+d x} \left (a^3 \left (8 c^2+14 c d x+3 d^2 x^2\right )+a^2 b x \left (-14 c^2-38 c d x+3 d^2 x^2\right )+5 a b^2 c x^2 (7 c-20 d x)+105 b^3 c^2 x^3\right )}{x^3 \sqrt{a+b x}}}{48 a^{9/2} c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a]*Sqrt[c]*Sqrt[c + d*x]*(105*b^3*c^2*x^3 + 5*a*b^2*c*x^2*(7*c - 20*d*
x) + a^2*b*x*(-14*c^2 - 38*c*d*x + 3*d^2*x^2) + a^3*(8*c^2 + 14*c*d*x + 3*d^2*x^
2)))/(x^3*Sqrt[a + b*x]) - 3*(-(b*c) + a*d)*(-35*b^2*c^2 + 10*a*b*c*d + a^2*d^2)
*Log[x] + 3*(-(b*c) + a*d)*(-35*b^2*c^2 + 10*a*b*c*d + 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^(9/2)*c^(3/2))

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Maple [B]  time = 0.045, size = 707, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.712647, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (8 \, a^{3} c^{2} +{\left (105 \, b^{3} c^{2} - 100 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3} +{\left (35 \, a b^{2} c^{2} - 38 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} x^{2} - 14 \,{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} +{\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{96 \,{\left (a^{4} b c x^{4} + a^{5} c x^{3}\right )} \sqrt{a c}}, -\frac{2 \,{\left (8 \, a^{3} c^{2} +{\left (105 \, b^{3} c^{2} - 100 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3} +{\left (35 \, a b^{2} c^{2} - 38 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} x^{2} - 14 \,{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} +{\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{48 \,{\left (a^{4} b c x^{4} + a^{5} c x^{3}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError