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

Optimal. Leaf size=189 \[ \frac{(5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+3 b^2 c^2\right )}{3 a c^3 \sqrt{c+d x} (b c-a d)^2}-\frac{d \sqrt{a+b x} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{a+b x}}{a c x (c+d x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.577128, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+3 b^2 c^2\right )}{3 a c^3 \sqrt{c+d x} (b c-a d)^2}-\frac{d \sqrt{a+b x} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{a+b x}}{a c x (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 66.389, size = 170, normalized size = 0.9 \[ - \frac{\sqrt{a + b x}}{a c x \left (c + d x\right )^{\frac{3}{2}}} - \frac{d \sqrt{a + b x} \left (5 a d - 3 b c\right )}{3 a c^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{d \sqrt{a + b x} \left (15 a^{2} d^{2} - 22 a b c d + 3 b^{2} c^{2}\right )}{3 a c^{3} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{\left (5 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.833418, size = 174, normalized size = 0.92 \[ \frac{-\frac{3 \log (x) (5 a d+b c)}{a^{3/2}}+\frac{3 (5 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{4 d^2 (4 b c-3 a d)}{(c+d x) (b c-a d)^2}+\frac{2 c d^2}{(c+d x)^2 (b c-a d)}-\frac{3}{a x}\right )}{6 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.052, size = 919, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x+a)^(1/2)/a/c^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^3*d^5-27*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^2*b*c*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^3*a*b^2*c^2*d^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*b^3*c^3*d^2+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^3*c*d^4-54*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^2*a^2*b*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^2*a*b^2*c^3*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*b^3*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*a^3*c^2*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*a^2*b*c^3*d^2+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^2*c^4*d+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*b^3*c^5-30*x^2*a^2*d^4*((b*x+a)*(d*x+c))^(1/
2)*(a*c)^(1/2)+44*x^2*a*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6*x^2*b^2*c^
2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-40*x*a^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)
*(a*c)^(1/2)+60*x*a*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-12*x*b^2*c^3*d
*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6*a^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)
^(1/2)+12*a*b*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6*b^2*c^4*((b*x+a)*(d*x+
c))^(1/2)*(a*c)^(1/2))/(a*d-b*c)^2/(a*c)^(1/2)/x/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)
^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.478795, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*(3*b^2*c^4 - 6*a*b*c^3*d + 3*a^2*c^2*d^2 + (3*b^2*c^2*d^2 - 22*a*b*c*d
^3 + 15*a^2*d^4)*x^2 + 2*(3*b^2*c^3*d - 15*a*b*c^2*d^2 + 10*a^2*c*d^3)*x)*sqrt(a
*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*((b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 - 9*a^2*b*c*
d^4 + 5*a^3*d^5)*x^3 + 2*(b^3*c^4*d + 3*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3 + 5*a^3*
c*d^4)*x^2 + (b^3*c^5 + 3*a*b^2*c^4*d - 9*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*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*b^2*c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^3 + 2*(a*b^2*c^6*d - 2*a^2
*b*c^5*d^2 + a^3*c^4*d^3)*x^2 + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*x)*sqr
t(a*c)), -1/6*(2*(3*b^2*c^4 - 6*a*b*c^3*d + 3*a^2*c^2*d^2 + (3*b^2*c^2*d^2 - 22*
a*b*c*d^3 + 15*a^2*d^4)*x^2 + 2*(3*b^2*c^3*d - 15*a*b*c^2*d^2 + 10*a^2*c*d^3)*x)
*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*((b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 - 9*
a^2*b*c*d^4 + 5*a^3*d^5)*x^3 + 2*(b^3*c^4*d + 3*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3
+ 5*a^3*c*d^4)*x^2 + (b^3*c^5 + 3*a*b^2*c^4*d - 9*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)
*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*b^2*c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^3 + 2*(a*b^2*c^6*d - 2
*a^2*b*c^5*d^2 + a^3*c^4*d^3)*x^2 + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*x)
*sqrt(-a*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*sqrt(a + b*x)*(c + d*x)**(5/2)), x)

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

[Out]

Exception raised: TypeError