3.585 \(\int \frac{\sqrt{a+b x}}{x^3 (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=234 \[ -\frac{d \sqrt{a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{12 a c^4 \sqrt{c+d x} (b c-a d)}+\frac{\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{9/2}}-\frac{d \sqrt{a+b x} (3 b c-35 a d)}{12 a c^3 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac{\sqrt{a+b x}}{2 c x^2 (c+d x)^{3/2}} \]
[Out]
-(d*(3*b*c - 35*a*d)*Sqrt[a + b*x])/(12*a*c^3*(c + d*x)^(3/2)) - Sqrt[a + b*x]/(
2*c*x^2*(c + d*x)^(3/2)) - ((b*c - 7*a*d)*Sqrt[a + b*x])/(4*a*c^2*x*(c + d*x)^(3
/2)) - (d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a + b*x])/(12*a*c^4*(b*c
- a*d)*Sqrt[c + d*x]) + ((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(9/2))
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Rubi [A] time = 0.783347, antiderivative size = 234, 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
\[ -\frac{d \sqrt{a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{12 a c^4 \sqrt{c+d x} (b c-a d)}+\frac{\left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{9/2}}-\frac{d \sqrt{a+b x} (3 b c-35 a d)}{12 a c^3 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}-\frac{\sqrt{a+b x}}{2 c x^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/(x^3*(c + d*x)^(5/2)),x]
[Out]
-(d*(3*b*c - 35*a*d)*Sqrt[a + b*x])/(12*a*c^3*(c + d*x)^(3/2)) - Sqrt[a + b*x]/(
2*c*x^2*(c + d*x)^(3/2)) - ((b*c - 7*a*d)*Sqrt[a + b*x])/(4*a*c^2*x*(c + d*x)^(3
/2)) - (d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a + b*x])/(12*a*c^4*(b*c
- a*d)*Sqrt[c + d*x]) + ((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(9/2))
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Rubi in Sympy [A] time = 114.79, size = 214, normalized size = 0.91 \[ - \frac{\sqrt{a + b x}}{2 c x^{2} \left (c + d x\right )^{\frac{3}{2}}} + \frac{\sqrt{a + b x} \left (7 a d - b c\right )}{4 a c^{2} x \left (c + d x\right )^{\frac{3}{2}}} + \frac{d \sqrt{a + b x} \left (35 a d - 3 b c\right )}{12 a c^{3} \left (c + d x\right )^{\frac{3}{2}}} + \frac{d \sqrt{a + b x} \left (105 a^{2} d^{2} - 100 a b c d + 3 b^{2} c^{2}\right )}{12 a c^{4} \sqrt{c + d x} \left (a d - b c\right )} - \frac{\left (35 a^{2} d^{2} - 10 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(5/2),x)
[Out]
-sqrt(a + b*x)/(2*c*x**2*(c + d*x)**(3/2)) + sqrt(a + b*x)*(7*a*d - b*c)/(4*a*c*
*2*x*(c + d*x)**(3/2)) + d*sqrt(a + b*x)*(35*a*d - 3*b*c)/(12*a*c**3*(c + d*x)**
(3/2)) + d*sqrt(a + b*x)*(105*a**2*d**2 - 100*a*b*c*d + 3*b**2*c**2)/(12*a*c**4*
sqrt(c + d*x)*(a*d - b*c)) - (35*a**2*d**2 - 10*a*b*c*d - b**2*c**2)*atanh(sqrt(
c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(4*a**(3/2)*c**(9/2))
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Mathematica [A] time = 0.857534, size = 206, normalized size = 0.88 \[ \frac{-\frac{3 \log (x) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )}{a^{3/2}}+\frac{3 \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{8 d^2 (8 b c-9 a d)}{(c+d x) (b c-a d)}+\frac{33 a d-3 b c}{a x}+\frac{8 c d^2}{(c+d x)^2}-\frac{6 c}{x^2}\right )}{24 c^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/(x^3*(c + d*x)^(5/2)),x]
[Out]
(2*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*((-6*c)/x^2 + (-3*b*c + 33*a*d)/(a*x) + (
8*c*d^2)/(c + d*x)^2 + (8*d^2*(8*b*c - 9*a*d))/((b*c - a*d)*(c + d*x))) - (3*(b^
2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Log[x])/a^(3/2) + (3*(b^2*c^2 + 10*a*b*c*d - 35
*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]])/a^(3/2))/(24*c^(9/2))
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Maple [B] time = 0.046, size = 988, normalized size = 4.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)/x^3/(d*x+c)^(5/2),x)
[Out]
-1/24*(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^5-135*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^4+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^
4*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^4*b^3*c^3*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^4-270*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^3+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*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^3*b^3*c^4*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^3-135*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^2+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^2*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^2*b^3*c^5-210*x^3*a^2*d^4*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)+200*x^3*a*b*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x^3*b^2*c^
2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*x^2*a^2*c*d^3*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+276*x^2*a*b*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-12*x^2*b
^2*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*x*a^2*c^2*d^2*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2)+48*x*a*b*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x*b^2*c^4
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^2*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)-12*a*b*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c^4/a*(b*x+a)^(1/2)/(a*d-b
*c)/(a*c)^(1/2)/x^2/((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. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/((d*x + c)^(5/2)*x^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.799214, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/((d*x + c)^(5/2)*x^3),x, algorithm="fricas")
[Out]
[-1/48*(4*(6*a*b*c^4 - 6*a^2*c^3*d + (3*b^2*c^2*d^2 - 100*a*b*c*d^3 + 105*a^2*d^
4)*x^3 + 2*(3*b^2*c^3*d - 69*a*b*c^2*d^2 + 70*a^2*c*d^3)*x^2 + 3*(b^2*c^4 - 8*a*
b*c^3*d + 7*a^2*c^2*d^2)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 3*((b^3*c^3*
d^2 + 9*a*b^2*c^2*d^3 - 45*a^2*b*c*d^4 + 35*a^3*d^5)*x^4 + 2*(b^3*c^4*d + 9*a*b^
2*c^3*d^2 - 45*a^2*b*c^2*d^3 + 35*a^3*c*d^4)*x^3 + (b^3*c^5 + 9*a*b^2*c^4*d - 45
*a^2*b*c^3*d^2 + 35*a^3*c^2*d^3)*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*b*c^5*d^2 - a^2*c^4*d^3)*x^4 +
2*(a*b*c^6*d - a^2*c^5*d^2)*x^3 + (a*b*c^7 - a^2*c^6*d)*x^2)*sqrt(a*c)), -1/24*
(2*(6*a*b*c^4 - 6*a^2*c^3*d + (3*b^2*c^2*d^2 - 100*a*b*c*d^3 + 105*a^2*d^4)*x^3
+ 2*(3*b^2*c^3*d - 69*a*b*c^2*d^2 + 70*a^2*c*d^3)*x^2 + 3*(b^2*c^4 - 8*a*b*c^3*d
+ 7*a^2*c^2*d^2)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*((b^3*c^3*d^2 +
9*a*b^2*c^2*d^3 - 45*a^2*b*c*d^4 + 35*a^3*d^5)*x^4 + 2*(b^3*c^4*d + 9*a*b^2*c^3*
d^2 - 45*a^2*b*c^2*d^3 + 35*a^3*c*d^4)*x^3 + (b^3*c^5 + 9*a*b^2*c^4*d - 45*a^2*b
*c^3*d^2 + 35*a^3*c^2*d^3)*x^2)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(s
qrt(b*x + a)*sqrt(d*x + c)*a*c)))/(((a*b*c^5*d^2 - a^2*c^4*d^3)*x^4 + 2*(a*b*c^6
*d - a^2*c^5*d^2)*x^3 + (a*b*c^7 - a^2*c^6*d)*x^2)*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((b*x+a)**(1/2)/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(sqrt(b*x + a)/((d*x + c)^(5/2)*x^3),x, algorithm="giac")
[Out]
Exception raised: TypeError