3.484 \(\int \frac{1}{x^3 \left (a+b \sqrt{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=306 \[ \frac{a b^2 d^2 \left (a^2+11 b^2 c\right )}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )}-\frac{a-b \sqrt{c+d x}}{2 x^2 \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}-\frac{b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt{c+d x}\right )}{2 c x \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )}+\frac{b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac{2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac{a b d^2 \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4} \]

[Out]

(a*b^2*(a^2 + 11*b^2*c)*d^2)/(2*c*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])) - (a -
b*Sqrt[c + d*x])/(2*(a^2 - b^2*c)*x^2*(a + b*Sqrt[c + d*x])) - (b*d*(3*a*b*c - (
a^2 + 2*b^2*c)*Sqrt[c + d*x]))/(2*c*(a^2 - b^2*c)^2*x*(a + b*Sqrt[c + d*x])) - (
a*b*(a^4 - 10*a^2*b^2*c - 15*b^4*c^2)*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*c^(
3/2)*(a^2 - b^2*c)^4) + (b^4*(5*a^2 + b^2*c)*d^2*Log[x])/(a^2 - b^2*c)^4 - (2*b^
4*(5*a^2 + b^2*c)*d^2*Log[a + b*Sqrt[c + d*x]])/(a^2 - b^2*c)^4

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Rubi [A]  time = 0.837213, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{a b^2 d^2 \left (a^2+11 b^2 c\right )}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )}-\frac{a-b \sqrt{c+d x}}{2 x^2 \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}-\frac{b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt{c+d x}\right )}{2 c x \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )}+\frac{b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac{2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac{a b d^2 \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

(a*b^2*(a^2 + 11*b^2*c)*d^2)/(2*c*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])) - (a -
b*Sqrt[c + d*x])/(2*(a^2 - b^2*c)*x^2*(a + b*Sqrt[c + d*x])) - (b*d*(3*a*b*c - (
a^2 + 2*b^2*c)*Sqrt[c + d*x]))/(2*c*(a^2 - b^2*c)^2*x*(a + b*Sqrt[c + d*x])) - (
a*b*(a^4 - 10*a^2*b^2*c - 15*b^4*c^2)*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*c^(
3/2)*(a^2 - b^2*c)^4) + (b^4*(5*a^2 + b^2*c)*d^2*Log[x])/(a^2 - b^2*c)^4 - (2*b^
4*(5*a^2 + b^2*c)*d^2*Log[a + b*Sqrt[c + d*x]])/(a^2 - b^2*c)^4

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Rubi in Sympy [A]  time = 71.6826, size = 275, normalized size = 0.9 \[ \frac{a b^{2} d^{2} \left (a^{2} + 11 b^{2} c\right )}{2 c \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )^{3}} - \frac{a b d^{2} \left (a^{4} - 10 a^{2} b^{2} c - 15 b^{4} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )^{4}} + \frac{b^{4} d^{2} \left (5 a^{2} + b^{2} c\right ) \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{4}} - \frac{2 b^{4} d^{2} \left (5 a^{2} + b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{4}} + \frac{b d \left (- 6 a b c + \left (2 a^{2} + 4 b^{2} c\right ) \sqrt{c + d x}\right )}{4 c x \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )^{2}} - \frac{a - b \sqrt{c + d x}}{2 x^{2} \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*b**2*d**2*(a**2 + 11*b**2*c)/(2*c*(a + b*sqrt(c + d*x))*(a**2 - b**2*c)**3) -
a*b*d**2*(a**4 - 10*a**2*b**2*c - 15*b**4*c**2)*atanh(sqrt(c + d*x)/sqrt(c))/(2*
c**(3/2)*(a**2 - b**2*c)**4) + b**4*d**2*(5*a**2 + b**2*c)*log(-d*x)/(a**2 - b**
2*c)**4 - 2*b**4*d**2*(5*a**2 + b**2*c)*log(a + b*sqrt(c + d*x))/(a**2 - b**2*c)
**4 + b*d*(-6*a*b*c + (2*a**2 + 4*b**2*c)*sqrt(c + d*x))/(4*c*x*(a + b*sqrt(c +
d*x))*(a**2 - b**2*c)**2) - (a - b*sqrt(c + d*x))/(2*x**2*(a + b*sqrt(c + d*x))*
(a**2 - b**2*c))

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Mathematica [A]  time = 0.616692, size = 390, normalized size = 1.27 \[ \frac{1}{2} \left (-\frac{2 b^2 d \left (3 a^2+b^2 c\right )}{x \left (a^2-b^2 c\right )^3}-\frac{a^2+b^2 c}{x^2 \left (a^2-b^2 c\right )^2}+\frac{4 a^2 b^4 d^2}{\left (a^2-b^2 c\right )^3 \left (a^2-b^2 (c+d x)\right )}+\frac{2 b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac{2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a^2-b^2 (c+d x)\right )}{\left (a^2-b^2 c\right )^4}-\frac{4 b^4 d^2 \left (5 a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )}{\left (a^2-b^2 c\right )^4}+\frac{a \sqrt{c+d x} \left (a^4 b (2 c+d x)-a^2 b^3 (d x-2 c)^2+b^5 c \left (2 c^2-5 c d x-11 d^2 x^2\right )\right )}{c x^2 \left (b^2 c-a^2\right )^3 \left (b^2 (c+d x)-a^2\right )}+\frac{d^2 \left (a^5 (-b)+10 a^3 b^3 c+15 a b^5 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2} \left (a^2-b^2 c\right )^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((a^2 + b^2*c)/((a^2 - b^2*c)^2*x^2)) - (2*b^2*(3*a^2 + b^2*c)*d)/((a^2 - b^2*
c)^3*x) + (4*a^2*b^4*d^2)/((a^2 - b^2*c)^3*(a^2 - b^2*(c + d*x))) + (a*Sqrt[c +
d*x]*(-(a^2*b^3*(-2*c + d*x)^2) + a^4*b*(2*c + d*x) + b^5*c*(2*c^2 - 5*c*d*x - 1
1*d^2*x^2)))/(c*(-a^2 + b^2*c)^3*x^2*(-a^2 + b^2*(c + d*x))) - (4*b^4*(5*a^2 + b
^2*c)*d^2*ArcTanh[(b*Sqrt[c + d*x])/a])/(a^2 - b^2*c)^4 + ((-(a^5*b) + 10*a^3*b^
3*c + 15*a*b^5*c^2)*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(c^(3/2)*(a^2 - b^2*c)^4
) + (2*b^4*(5*a^2 + b^2*c)*d^2*Log[x])/(a^2 - b^2*c)^4 - (2*b^4*(5*a^2 + b^2*c)*
d^2*Log[a^2 - b^2*(c + d*x)])/(a^2 - b^2*c)^4)/2

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Maple [B]  time = 0.027, size = 612, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/2/(-b^2*c+a^2)^4/x^2*a*b^5*c*(d*x+c)^(3/2)+3/(-b^2*c+a^2)^4/x^2*a^3*b^3*(d*x+
c)^(3/2)+1/2/(-b^2*c+a^2)^4/x^2*a^5*b/c*(d*x+c)^(3/2)+d/(-b^2*c+a^2)^4/x*b^6*c^2
-1/2/(-b^2*c+a^2)^4/x^2*b^6*c^3+2*d/(-b^2*c+a^2)^4/x*a^2*b^4*c+1/2/(-b^2*c+a^2)^
4/x^2*a^2*b^4*c^2-3*d/(-b^2*c+a^2)^4/x*a^4*b^2+1/2/(-b^2*c+a^2)^4/x^2*a^4*b^2*c+
9/2/(-b^2*c+a^2)^4/x^2*(d*x+c)^(1/2)*a*b^5*c^2-5/(-b^2*c+a^2)^4/x^2*(d*x+c)^(1/2
)*a^3*b^3*c+1/2/(-b^2*c+a^2)^4/x^2*(d*x+c)^(1/2)*b*a^5-1/2/(-b^2*c+a^2)^4/x^2*a^
6+d^2/(-b^2*c+a^2)^4*b^6*c*ln(c*d*x)+5*d^2/(-b^2*c+a^2)^4*b^4*ln(c*d*x)*a^2+15/2
*d^2/(-b^2*c+a^2)^4*b^5*c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*a+5*d^2/(-b^2*c+a
^2)^4*b^3/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*a^3-1/2*d^2/(-b^2*c+a^2)^4*b/c^
(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*a^5+2*d^2*b^4/(-b^2*c+a^2)^3*a/(a+b*(d*x+c)
^(1/2))-2*d^2*b^6/(-b^2*c+a^2)^4*ln(a+b*(d*x+c)^(1/2))*c-10*d^2*b^4/(-b^2*c+a^2)
^4*ln(a+b*(d*x+c)^(1/2))*a^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/((sqrt(d*x + c)*b + a)^2*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60741, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*(b^7*c^4 - 3*a^2*b^5*c^3 + 3*a^4*b^3*c^2 - a^6*b*c - 2*(b^7*c^2 + 5*a^2
*b^5*c)*d^2*x^2*log(x) - (2*b^7*c^3 - 3*a^2*b^5*c^2 + a^6*b)*d*x)*sqrt(d*x + c)*
sqrt(c) + 8*((b^7*c^2 + 5*a^2*b^5*c)*sqrt(d*x + c)*sqrt(c)*d^2*x^2 + (a*b^6*c^2
+ 5*a^3*b^4*c)*sqrt(c)*d^2*x^2)*log(sqrt(d*x + c)*b + a) - ((15*a*b^6*c^2 + 10*a
^3*b^4*c - a^5*b^2)*sqrt(d*x + c)*d^2*x^2 + (15*a^2*b^5*c^2 + 10*a^4*b^3*c - a^6
*b)*d^2*x^2)*log(((d*x + 2*c)*sqrt(c) + 2*sqrt(d*x + c)*c)/x) - 2*(a*b^6*c^4 - 3
*a^3*b^4*c^3 + 3*a^5*b^2*c^2 - a^7*c + 2*(a*b^6*c^2 + 5*a^3*b^4*c)*d^2*x^2*log(x
) - (11*a*b^6*c^2 - 10*a^3*b^4*c - a^5*b^2)*d^2*x^2 - 3*(a*b^6*c^3 - 2*a^3*b^4*c
^2 + a^5*b^2*c)*d*x)*sqrt(c))/((b^9*c^5 - 4*a^2*b^7*c^4 + 6*a^4*b^5*c^3 - 4*a^6*
b^3*c^2 + a^8*b*c)*sqrt(d*x + c)*sqrt(c)*x^2 + (a*b^8*c^5 - 4*a^3*b^6*c^4 + 6*a^
5*b^4*c^3 - 4*a^7*b^2*c^2 + a^9*c)*sqrt(c)*x^2), -1/2*((b^7*c^4 - 3*a^2*b^5*c^3
+ 3*a^4*b^3*c^2 - a^6*b*c - 2*(b^7*c^2 + 5*a^2*b^5*c)*d^2*x^2*log(x) - (2*b^7*c^
3 - 3*a^2*b^5*c^2 + a^6*b)*d*x)*sqrt(d*x + c)*sqrt(-c) + ((15*a*b^6*c^2 + 10*a^3
*b^4*c - a^5*b^2)*sqrt(d*x + c)*d^2*x^2 + (15*a^2*b^5*c^2 + 10*a^4*b^3*c - a^6*b
)*d^2*x^2)*arctan(c/(sqrt(d*x + c)*sqrt(-c))) + 4*((b^7*c^2 + 5*a^2*b^5*c)*sqrt(
d*x + c)*sqrt(-c)*d^2*x^2 + (a*b^6*c^2 + 5*a^3*b^4*c)*sqrt(-c)*d^2*x^2)*log(sqrt
(d*x + c)*b + a) - (a*b^6*c^4 - 3*a^3*b^4*c^3 + 3*a^5*b^2*c^2 - a^7*c + 2*(a*b^6
*c^2 + 5*a^3*b^4*c)*d^2*x^2*log(x) - (11*a*b^6*c^2 - 10*a^3*b^4*c - a^5*b^2)*d^2
*x^2 - 3*(a*b^6*c^3 - 2*a^3*b^4*c^2 + a^5*b^2*c)*d*x)*sqrt(-c))/((b^9*c^5 - 4*a^
2*b^7*c^4 + 6*a^4*b^5*c^3 - 4*a^6*b^3*c^2 + a^8*b*c)*sqrt(d*x + c)*sqrt(-c)*x^2
+ (a*b^8*c^5 - 4*a^3*b^6*c^4 + 6*a^5*b^4*c^3 - 4*a^7*b^2*c^2 + a^9*c)*sqrt(-c)*x
^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef