3.779 \(\int \frac{1}{x^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=264 \[ \frac{(5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{7/2}}-\frac{d \sqrt{a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{b (3 b c-a d)}{a^2 c \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{d \sqrt{a+b x} \left (-15 a^3 d^3+31 a^2 b c d^2-9 a b^2 c^2 d+9 b^3 c^3\right )}{3 a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}} \]

[Out]

-((b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))) - 1/(a*c*
x*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (d*(9*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[a
+ b*x])/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (d*(9*b^3*c^3 - 9*a*b^2*c^2
*d + 31*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x])/(3*a^2*c^3*(b*c - a*d)^3*Sqrt[c
+ d*x]) + ((3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*
x])])/(a^(5/2)*c^(7/2))

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Rubi [A]  time = 0.885541, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{7/2}}-\frac{d \sqrt{a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{b (3 b c-a d)}{a^2 c \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{d \sqrt{a+b x} \left (-15 a^3 d^3+31 a^2 b c d^2-9 a b^2 c^2 d+9 b^3 c^3\right )}{3 a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{1}{a c x \sqrt{a+b x} (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

-((b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))) - 1/(a*c*
x*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (d*(9*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[a
+ b*x])/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (d*(9*b^3*c^3 - 9*a*b^2*c^2
*d + 31*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x])/(3*a^2*c^3*(b*c - a*d)^3*Sqrt[c
+ d*x]) + ((3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*
x])])/(a^(5/2)*c^(7/2))

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Rubi in Sympy [A]  time = 114.612, size = 248, normalized size = 0.94 \[ - \frac{1}{a c x \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}} - \frac{b \left (a d - 3 b c\right )}{a^{2} c \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{d \sqrt{a + b x} \left (5 a^{2} d^{2} - 6 a b c d + 9 b^{2} c^{2}\right )}{3 a^{2} c^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{d \sqrt{a + b x} \left (15 a^{3} d^{3} - 31 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 9 b^{3} c^{3}\right )}{3 a^{2} c^{3} \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{\left (5 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

-1/(a*c*x*sqrt(a + b*x)*(c + d*x)**(3/2)) - b*(a*d - 3*b*c)/(a**2*c*sqrt(a + b*x
)*(c + d*x)**(3/2)*(a*d - b*c)) - d*sqrt(a + b*x)*(5*a**2*d**2 - 6*a*b*c*d + 9*b
**2*c**2)/(3*a**2*c**2*(c + d*x)**(3/2)*(a*d - b*c)**2) - d*sqrt(a + b*x)*(15*a*
*3*d**3 - 31*a**2*b*c*d**2 + 9*a*b**2*c**2*d - 9*b**3*c**3)/(3*a**2*c**3*sqrt(c
+ d*x)*(a*d - b*c)**3) + (5*a*d + 3*b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sq
rt(c + d*x)))/(a**(5/2)*c**(7/2))

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Mathematica [A]  time = 1.19871, size = 212, normalized size = 0.8 \[ -\frac{\log (x) (5 a d+3 b c)}{2 a^{5/2} c^{7/2}}+\frac{(5 a d+3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{5/2} c^{7/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 b^4}{a^2 (a+b x) (a d-b c)^3}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (11 b c-6 a d)}{3 c^3 (c+d x) (b c-a d)^3}-\frac{2 d^3}{3 c^2 (c+d x)^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*(-(1/(a^2*c^3*x)) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a
+ b*x)) - (2*d^3)/(3*c^2*(b*c - a*d)^2*(c + d*x)^2) - (2*d^3*(11*b*c - 6*a*d))/
(3*c^3*(b*c - a*d)^3*(c + d*x))) - ((3*b*c + 5*a*d)*Log[x])/(2*a^(5/2)*c^(7/2))
+ ((3*b*c + 5*a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*S
qrt[c + d*x]])/(2*a^(5/2)*c^(7/2))

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Maple [B]  time = 0.072, size = 1692, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)

[Out]

1/6*(-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*b^5
*c^5*d+30*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^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*a^5
*c^2*d^4-9*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*a*b
^4*c^6-30*x^2*a^4*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+18*x*b^4*c^5*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)-6*a^4*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+6*a
*b^3*c^5*(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^6+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*a^2*b^3*c^5*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^3*a^5*d^6-9*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*b^5*c^6-9*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^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^2*a*b^4*c^5*d-36*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*a^4*b*c^3*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*a^3*b^2*c^4*d^2+36*x^2*b^4*c^4*d*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)-40*x*a^4*c*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)+18*a^3*b*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-18*a^2*b^2*c^4*d*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)-36*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^5-30*x^3*a^3*b*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)+18*x^3*b^4*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+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^4*a^2*b^3*c^2*d^4+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^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^3*a^4*b*c*d^5-54*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^
4+48*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^3+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*b^4*c^4*d^2-57*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*b*c^2*d^4+42*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^2+62*x^3*a^2*b^2*c*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)-18*x^3*a*b^3*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+22*x^2*a^3*b*c*d^4*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+66*x^2*a^2*b^2*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)-30*x^2*a*b^3*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+78*x*a^3*
b*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-18*x*a^2*b^2*c^3*d^2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)-6*x*a*b^3*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c^3/
a^2/x/(a*c)^(1/2)/(a*d-b*c)^3/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(1/2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^2), x)

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Fricas [A]  time = 0.70761, 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)^(5/2)*x^2),x, algorithm="fricas")

[Out]

[-1/12*(4*(3*a*b^3*c^5 - 9*a^2*b^2*c^4*d + 9*a^3*b*c^3*d^2 - 3*a^4*c^2*d^3 + (9*
b^4*c^3*d^2 - 9*a*b^3*c^2*d^3 + 31*a^2*b^2*c*d^4 - 15*a^3*b*d^5)*x^3 + (18*b^4*c
^4*d - 15*a*b^3*c^3*d^2 + 33*a^2*b^2*c^2*d^3 + 11*a^3*b*c*d^4 - 15*a^4*d^5)*x^2
+ (9*b^4*c^5 - 3*a*b^3*c^4*d - 9*a^2*b^2*c^3*d^2 + 39*a^3*b*c^2*d^3 - 20*a^4*c*d
^4)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*((3*b^5*c^4*d^2 - 4*a*b^4*c^3*d
^3 - 6*a^2*b^3*c^2*d^4 + 12*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (6*b^5*c^5*d - 5*
a*b^4*c^4*d^2 - 16*a^2*b^3*c^3*d^3 + 18*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - 5*a^5*
d^6)*x^3 + (3*b^5*c^6 + 2*a*b^4*c^5*d - 14*a^2*b^3*c^4*d^2 + 19*a^4*b*c^2*d^4 -
10*a^5*c*d^5)*x^2 + (3*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 12*a^4*
b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*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^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a
^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^4 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a
^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^3 + (a^2*b^4*c^8 - a^3*b^3*c^7*d
- 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2 + (a^3*b^3*c^8 - 3*a
^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3)*x)*sqrt(a*c)), -1/6*(2*(3*a*b^3*c^
5 - 9*a^2*b^2*c^4*d + 9*a^3*b*c^3*d^2 - 3*a^4*c^2*d^3 + (9*b^4*c^3*d^2 - 9*a*b^3
*c^2*d^3 + 31*a^2*b^2*c*d^4 - 15*a^3*b*d^5)*x^3 + (18*b^4*c^4*d - 15*a*b^3*c^3*d
^2 + 33*a^2*b^2*c^2*d^3 + 11*a^3*b*c*d^4 - 15*a^4*d^5)*x^2 + (9*b^4*c^5 - 3*a*b^
3*c^4*d - 9*a^2*b^2*c^3*d^2 + 39*a^3*b*c^2*d^3 - 20*a^4*c*d^4)*x)*sqrt(-a*c)*sqr
t(b*x + a)*sqrt(d*x + c) - 3*((3*b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 - 6*a^2*b^3*c^2*d
^4 + 12*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (6*b^5*c^5*d - 5*a*b^4*c^4*d^2 - 16*a
^2*b^3*c^3*d^3 + 18*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (3*b^5*c^
6 + 2*a*b^4*c^5*d - 14*a^2*b^3*c^4*d^2 + 19*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 +
(3*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 12*a^4*b*c^3*d^3 - 5*a^5*c^
2*d^4)*x)*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^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c
^3*d^5)*x^4 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c
^4*d^4 - a^6*c^3*d^5)*x^3 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5
*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c
^6*d^2 - a^6*c^5*d^3)*x)*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(1/x**2/(b*x+a)**(3/2)/(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(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError