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

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

[Out]

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

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Rubi [A]  time = 0.412535, 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 \[ -\frac{5 \sqrt{a} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{9/2}}+\frac{5 \sqrt{a+b x} (3 b c-7 a d) (b c-a d)}{4 c^4 \sqrt{c+d x}}+\frac{5 (a+b x)^{3/2} (3 b c-7 a d) (b c-a d)}{12 a c^3 (c+d x)^{3/2}}-\frac{(a+b x)^{5/2} (3 b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.6606, size = 207, normalized size = 0.94 \[ - \frac{5 \sqrt{a} \left (a d - b c\right ) \left (7 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 c^{\frac{9}{2}}} + \frac{5 a \sqrt{a + b x} \sqrt{c + d x} \left (7 a d - 3 b c\right )}{4 c^{4} x} + \frac{2 d \left (a + b x\right )^{\frac{7}{2}}}{3 c x^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (7 a d - 3 b c\right )}{6 c^{2} x^{2} \sqrt{c + d x} \left (a d - b c\right )} - \frac{5 \left (a + b x\right )^{\frac{3}{2}} \left (7 a d - 3 b c\right )}{6 c^{3} x \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.3584, size = 211, normalized size = 0.96 \[ \frac{\frac{2 \sqrt{c} \sqrt{a+b x} \left (a^2 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )-a b c x \left (27 c^2+158 c d x+115 d^2 x^2\right )+8 b^2 c^2 x^2 (3 c+2 d x)\right )}{x^2 (c+d x)^{3/2}}+15 \sqrt{a} \log (x) (a d-b c) (7 a d-3 b c)-15 \sqrt{a} (a d-b c) (7 a d-3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{24 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*Sqrt[a + b*x]*(8*b^2*c^2*x^2*(3*c + 2*d*x) - a*b*c*x*(27*c^2 + 158*c
*d*x + 115*d^2*x^2) + a^2*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3)))/
(x^2*(c + d*x)^(3/2)) + 15*Sqrt[a]*(-(b*c) + a*d)*(-3*b*c + 7*a*d)*Log[x] - 15*S
qrt[a]*(-(b*c) + a*d)*(-3*b*c + 7*a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqr
t[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(24*c^(9/2))

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Maple [B]  time = 0.046, size = 758, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.892808, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (6 \, a^{2} c^{3} -{\left (16 \, b^{2} c^{2} d - 115 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} - 2 \,{\left (12 \, b^{2} c^{3} - 79 \, a b c^{2} d + 70 \, a^{2} c d^{2}\right )} x^{2} + 3 \,{\left (9 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )}}, -\frac{15 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) + 2 \,{\left (6 \, a^{2} c^{3} -{\left (16 \, b^{2} c^{2} d - 115 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} - 2 \,{\left (12 \, b^{2} c^{3} - 79 \, a b c^{2} d + 70 \, a^{2} c d^{2}\right )} x^{2} + 3 \,{\left (9 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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**3/(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^3),x, algorithm="giac")

[Out]

Exception raised: TypeError