3.771 \(\int \frac{1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=268 \[ \frac{5 (a d+b c)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}-\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{7/2}}+\frac{b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{4 a^3 c^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right ) (a d+b c)}{4 a^3 c^3 \sqrt{c+d x} (b c-a d)^2}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}} \]

[Out]

(b*(15*b^2*c^2 - 2*a*b*c*d - 5*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*Sq
rt[c + d*x]) - 1/(2*a*c*x^2*Sqrt[a + b*x]*Sqrt[c + d*x]) + (5*(b*c + a*d))/(4*a^
2*c^2*x*Sqrt[a + b*x]*Sqrt[c + d*x]) + (d*(b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d +
15*a^2*d^2)*Sqrt[a + b*x])/(4*a^3*c^3*(b*c - a*d)^2*Sqrt[c + d*x]) - (3*(5*b^2*
c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d
*x])])/(4*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.880872, antiderivative size = 268, 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 \[ \frac{5 (a d+b c)}{4 a^2 c^2 x \sqrt{a+b x} \sqrt{c+d x}}-\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{7/2}}+\frac{b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{4 a^3 c^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right ) (a d+b c)}{4 a^3 c^3 \sqrt{c+d x} (b c-a d)^2}-\frac{1}{2 a c x^2 \sqrt{a+b x} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(b*(15*b^2*c^2 - 2*a*b*c*d - 5*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*Sq
rt[c + d*x]) - 1/(2*a*c*x^2*Sqrt[a + b*x]*Sqrt[c + d*x]) + (5*(b*c + a*d))/(4*a^
2*c^2*x*Sqrt[a + b*x]*Sqrt[c + d*x]) + (d*(b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d +
15*a^2*d^2)*Sqrt[a + b*x])/(4*a^3*c^3*(b*c - a*d)^2*Sqrt[c + d*x]) - (3*(5*b^2*
c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d
*x])])/(4*a^(7/2)*c^(7/2))

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Rubi in Sympy [A]  time = 108.238, size = 252, normalized size = 0.94 \[ - \frac{1}{2 a c x^{2} \sqrt{a + b x} \sqrt{c + d x}} + \frac{5 \left (a d + b c\right )}{4 a^{2} c^{2} x \sqrt{a + b x} \sqrt{c + d x}} + \frac{b \left (5 a^{2} d^{2} + 2 a b c d - 15 b^{2} c^{2}\right )}{4 a^{3} c^{2} \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} + \frac{d \sqrt{a + b x} \left (a d + b c\right ) \left (15 a^{2} d^{2} - 22 a b c d + 15 b^{2} c^{2}\right )}{4 a^{3} c^{3} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{3 \left (4 a b c d - 5 \left (a d + b c\right )^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{7}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

-1/(2*a*c*x**2*sqrt(a + b*x)*sqrt(c + d*x)) + 5*(a*d + b*c)/(4*a**2*c**2*x*sqrt(
a + b*x)*sqrt(c + d*x)) + b*(5*a**2*d**2 + 2*a*b*c*d - 15*b**2*c**2)/(4*a**3*c**
2*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)) + d*sqrt(a + b*x)*(a*d + b*c)*(15*a**
2*d**2 - 22*a*b*c*d + 15*b**2*c**2)/(4*a**3*c**3*sqrt(c + d*x)*(a*d - b*c)**2) +
3*(4*a*b*c*d - 5*(a*d + b*c)**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c +
d*x)))/(4*a**(7/2)*c**(7/2))

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Mathematica [A]  time = 1.2353, size = 224, normalized size = 0.84 \[ \frac{3 \log (x) \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right )}{8 a^{7/2} c^{7/2}}-\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{7/2} c^{7/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 b^4}{a^3 (a+b x) (a d-b c)^2}+\frac{7 (a d+b c)}{4 a^3 c^3 x}-\frac{1}{2 a^2 c^2 x^2}+\frac{2 d^4}{c^3 (c+d x) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*(-1/(2*a^2*c^2*x^2) + (7*(b*c + a*d))/(4*a^3*c^3*x)
+ (2*b^4)/(a^3*(-(b*c) + a*d)^2*(a + b*x)) + (2*d^4)/(c^3*(b*c - a*d)^2*(c + d*x
))) + (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*Log[x])/(8*a^(7/2)*c^(7/2)) - (3*(5
*b^2*c^2 + 6*a*b*c*d + 5*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]])/(8*a^(7/2)*c^(7/2))

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Maple [B]  time = 0.063, size = 1372, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(3/2),x)

[Out]

-1/8/a^3/c^3*(-30*x^2*b^4*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a^4*c^2*d^2*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a^2*b^2*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)+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^4*a^4
*b*d^5+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^4*b^
5*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^2*a
^5*c*d^4+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^2*
a*b^4*c^5-12*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^2*c*d^4+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*a^5*d^5+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^5*c^5+14*x^3*a^2*b^2*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+14*x^3*a*b
^3*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*x^2*a^3*b*c*d^3*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+20*x^2*a^2*b^2*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
4*x^2*a*b^3*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10*x*a^3*b*c^2*d^2*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+10*x*a^2*b^2*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)-30*x^2*a^4*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10*x*a^4*c*d^3*(a*c)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)-10*x*a*b^3*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-8*a
^3*b*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*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^3*c^2*d^3-12*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^4*c^3*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^3*a^4*b*c*d^4-18*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^2*c^2*d^3-18*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^3*c^3*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^3*a*b^4*c^4*d-12*l
n((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*b*c^2*d^3
-6*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^2*c
^3*d^2-12*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^3*c^4*d-30*x^3*a^3*b*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^3*b^4*c^3*
d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/(a*d-
b*c)^2/x^2/(b*x+a)^(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(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.681651, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x^3),x, algorithm="fricas")

[Out]

[-1/16*(4*(2*a^2*b^2*c^4 - 4*a^3*b*c^3*d + 2*a^4*c^2*d^2 - (15*b^4*c^3*d - 7*a*b
^3*c^2*d^2 - 7*a^2*b^2*c*d^3 + 15*a^3*b*d^4)*x^3 - (15*b^4*c^4 - 2*a*b^3*c^3*d -
10*a^2*b^2*c^2*d^2 - 2*a^3*b*c*d^3 + 15*a^4*d^4)*x^2 - 5*(a*b^3*c^4 - a^2*b^2*c
^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(
(5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d
^5)*x^4 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4
*b*c*d^4 + 5*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 -
4*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x^2)*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^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2
+ a^5*b*c^3*d^3)*x^4 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d
^3)*x^3 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2)*sqrt(a*c)), -1/8*(2*(
2*a^2*b^2*c^4 - 4*a^3*b*c^3*d + 2*a^4*c^2*d^2 - (15*b^4*c^3*d - 7*a*b^3*c^2*d^2
- 7*a^2*b^2*c*d^3 + 15*a^3*b*d^4)*x^3 - (15*b^4*c^4 - 2*a*b^3*c^3*d - 10*a^2*b^2
*c^2*d^2 - 2*a^3*b*c*d^3 + 15*a^4*d^4)*x^2 - 5*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*
b*c^2*d^2 + a^4*c*d^3)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 3*((5*b^5*c^4
*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^4 +
(5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 +
5*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c
^2*d^3 + 5*a^5*c*d^4)*x^2)*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^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^
3)*x^4 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + (a^4*
b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2)*sqrt(-a*c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Integral(1/(x**3*(a + b*x)**(3/2)*(c + d*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 2.38019, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x^3),x, algorithm="giac")

[Out]

sage0*x