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

Optimal. Leaf size=220 \[ -\frac{5 \sqrt{c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt{a+b x}}+\frac{5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac{(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]

[Out]

(5*(7*b*c - 3*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(4*a^4*Sqrt[a + b*x]) + (5*(7*b*c
- 3*a*d)*(b*c - a*d)*(c + d*x)^(3/2))/(12*a^3*c*(a + b*x)^(3/2)) + ((7*b*c - 3*a
*d)*(c + d*x)^(5/2))/(4*a^2*c*x*(a + b*x)^(3/2)) - (c + d*x)^(7/2)/(2*a*c*x^2*(a
+ b*x)^(3/2)) - (5*Sqrt[c]*(7*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a
+ b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.40489, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 \sqrt{c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt{a+b x}}+\frac{5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac{(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac{(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(5*(7*b*c - 3*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(4*a^4*Sqrt[a + b*x]) + (5*(7*b*c
- 3*a*d)*(b*c - a*d)*(c + d*x)^(3/2))/(12*a^3*c*(a + b*x)^(3/2)) + ((7*b*c - 3*a
*d)*(c + d*x)^(5/2))/(4*a^2*c*x*(a + b*x)^(3/2)) - (c + d*x)^(7/2)/(2*a*c*x^2*(a
+ b*x)^(3/2)) - (5*Sqrt[c]*(7*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a
+ b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.4425, size = 207, normalized size = 0.94 \[ - \frac{2 b \left (c + d x\right )^{\frac{7}{2}}}{3 a x^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{\left (c + d x\right )^{\frac{5}{2}} \left (3 a d - 7 b c\right )}{6 a^{2} x^{2} \sqrt{a + b x} \left (a d - b c\right )} + \frac{5 \left (c + d x\right )^{\frac{3}{2}} \left (3 a d - 7 b c\right )}{6 a^{3} x \sqrt{a + b x}} - \frac{5 c \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{4 a^{4} x} - \frac{5 \sqrt{c} \left (a d - b c\right ) \left (3 a d - 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*(c + d*x)**(7/2)/(3*a*x**2*(a + b*x)**(3/2)*(a*d - b*c)) - (c + d*x)**(5/2)
*(3*a*d - 7*b*c)/(6*a**2*x**2*sqrt(a + b*x)*(a*d - b*c)) + 5*(c + d*x)**(3/2)*(3
*a*d - 7*b*c)/(6*a**3*x*sqrt(a + b*x)) - 5*c*sqrt(a + b*x)*sqrt(c + d*x)*(3*a*d
- 7*b*c)/(4*a**4*x) - 5*sqrt(c)*(a*d - b*c)*(3*a*d - 7*b*c)*atanh(sqrt(c)*sqrt(a
+ b*x)/(sqrt(a)*sqrt(c + d*x)))/(4*a**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.423981, size = 211, normalized size = 0.96 \[ \frac{\frac{2 \sqrt{a} \sqrt{c+d x} \left (-3 a^3 \left (2 c^2+9 c d x-8 d^2 x^2\right )+a^2 b x \left (21 c^2-158 c d x+16 d^2 x^2\right )+5 a b^2 c x^2 (28 c-23 d x)+105 b^3 c^2 x^3\right )}{x^2 (a+b x)^{3/2}}+15 \sqrt{c} \log (x) (7 b c-3 a d) (b c-a d)-15 \sqrt{c} (7 b c-3 a d) (b c-a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{24 a^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[c + d*x]*(105*b^3*c^2*x^3 + 5*a*b^2*c*x^2*(28*c - 23*d*x) - 3*a
^3*(2*c^2 + 9*c*d*x - 8*d^2*x^2) + a^2*b*x*(21*c^2 - 158*c*d*x + 16*d^2*x^2)))/(
x^2*(a + b*x)^(3/2)) + 15*Sqrt[c]*(7*b*c - 3*a*d)*(b*c - a*d)*Log[x] - 15*Sqrt[c
]*(7*b*c - 3*a*d)*(b*c - a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt
[a + b*x]*Sqrt[c + d*x]])/(24*a^(9/2))

_______________________________________________________________________________________

Maple [B]  time = 0.045, size = 758, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(d*x+c)^(1/2)*(45*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^2-150*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+90*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^2-300*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+210*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+45*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^2*a^4*c*d^2-150*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^2*a^3*b*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^2*a^2*b^2*c^3-32*x^3*a^2*b*d^2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+230*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)-48*x^2*a^3*d^2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+316*x^2*a^2*b*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*x^
2*a*b^2*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+54*x*a^3*c*d*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)-42*x*a^2*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^3*c^2*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/a^4/((b*x+a)*(d*x+c))^(1/2)/x^2/(a*c)^(1/2)/
(b*x+a)^(3/2)

_______________________________________________________________________________________

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)^(5/2)/((b*x + a)^(5/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.886089, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} +{\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (6 \, a^{3} c^{2} -{\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \,{\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \,{\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, -\frac{15 \,{\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} +{\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) + 2 \,{\left (6 \, a^{3} c^{2} -{\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \,{\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \,{\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*((7*b^4*c^2 - 10*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 + 2*(7*a*b^3*c^2 - 10*
a^2*b^2*c*d + 3*a^3*b*d^2)*x^3 + (7*a^2*b^2*c^2 - 10*a^3*b*c*d + 3*a^4*d^2)*x^2)
*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (
a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*
x)/x^2) - 4*(6*a^3*c^2 - (105*b^3*c^2 - 115*a*b^2*c*d + 16*a^2*b*d^2)*x^3 - 2*(7
0*a*b^2*c^2 - 79*a^2*b*c*d + 12*a^3*d^2)*x^2 - 3*(7*a^2*b*c^2 - 9*a^3*c*d)*x)*sq
rt(b*x + a)*sqrt(d*x + c))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2), -1/24*(15*((7*
b^4*c^2 - 10*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 + 2*(7*a*b^3*c^2 - 10*a^2*b^2*c*d +
3*a^3*b*d^2)*x^3 + (7*a^2*b^2*c^2 - 10*a^3*b*c*d + 3*a^4*d^2)*x^2)*sqrt(-c/a)*ar
ctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) + 2
*(6*a^3*c^2 - (105*b^3*c^2 - 115*a*b^2*c*d + 16*a^2*b*d^2)*x^3 - 2*(70*a*b^2*c^2
- 79*a^2*b*c*d + 12*a^3*d^2)*x^2 - 3*(7*a^2*b*c^2 - 9*a^3*c*d)*x)*sqrt(b*x + a)
*sqrt(d*x + c))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)]

_______________________________________________________________________________________

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)**(5/2)/x**3/(b*x+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(5/2)/((b*x + a)^(5/2)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError