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

Optimal. Leaf size=277 \[ \frac{d \sqrt{a+b x} \left (-35 a^2 d^2+18 a b c d+9 b^2 c^2\right )}{12 a^2 c^3 (c+d x)^{3/2} (b c-a d)}+\frac{\sqrt{a+b x} (7 a d+3 b c)}{4 a^2 c^2 x (c+d x)^{3/2}}-\frac{\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{9/2}}+\frac{d \sqrt{a+b x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{12 a^2 c^4 \sqrt{c+d x} (b c-a d)^2}-\frac{\sqrt{a+b x}}{2 a c x^2 (c+d x)^{3/2}} \]

[Out]

(d*(9*b^2*c^2 + 18*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x])/(12*a^2*c^3*(b*c - a*d)*
(c + d*x)^(3/2)) - Sqrt[a + b*x]/(2*a*c*x^2*(c + d*x)^(3/2)) + ((3*b*c + 7*a*d)*
Sqrt[a + b*x])/(4*a^2*c^2*x*(c + d*x)^(3/2)) + (d*(9*b^3*c^3 + 15*a*b^2*c^2*d -
145*a^2*b*c*d^2 + 105*a^3*d^3)*Sqrt[a + b*x])/(12*a^2*c^4*(b*c - a*d)^2*Sqrt[c +
d*x]) - ((3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(9/2))

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Rubi [A]  time = 0.860751, antiderivative size = 277, 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 \[ \frac{d \sqrt{a+b x} \left (-35 a^2 d^2+18 a b c d+9 b^2 c^2\right )}{12 a^2 c^3 (c+d x)^{3/2} (b c-a d)}+\frac{\sqrt{a+b x} (7 a d+3 b c)}{4 a^2 c^2 x (c+d x)^{3/2}}-\frac{\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{9/2}}+\frac{d \sqrt{a+b x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{12 a^2 c^4 \sqrt{c+d x} (b c-a d)^2}-\frac{\sqrt{a+b x}}{2 a c x^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d*(9*b^2*c^2 + 18*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x])/(12*a^2*c^3*(b*c - a*d)*
(c + d*x)^(3/2)) - Sqrt[a + b*x]/(2*a*c*x^2*(c + d*x)^(3/2)) + ((3*b*c + 7*a*d)*
Sqrt[a + b*x])/(4*a^2*c^2*x*(c + d*x)^(3/2)) + (d*(9*b^3*c^3 + 15*a*b^2*c^2*d -
145*a^2*b*c*d^2 + 105*a^3*d^3)*Sqrt[a + b*x])/(12*a^2*c^4*(b*c - a*d)^2*Sqrt[c +
d*x]) - ((3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(9/2))

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Rubi in Sympy [A]  time = 124.195, size = 264, normalized size = 0.95 \[ - \frac{\sqrt{a + b x}}{2 a c x^{2} \left (c + d x\right )^{\frac{3}{2}}} + \frac{\sqrt{a + b x} \left (7 a d + 3 b c\right )}{4 a^{2} c^{2} x \left (c + d x\right )^{\frac{3}{2}}} + \frac{d \sqrt{a + b x} \left (35 a^{2} d^{2} - 18 a b c d - 9 b^{2} c^{2}\right )}{12 a^{2} c^{3} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d \sqrt{a + b x} \left (105 a^{3} d^{3} - 145 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 9 b^{3} c^{3}\right )}{12 a^{2} c^{4} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{\left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{5}{2}} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(2*a*c*x**2*(c + d*x)**(3/2)) + sqrt(a + b*x)*(7*a*d + 3*b*c)/(4*
a**2*c**2*x*(c + d*x)**(3/2)) + d*sqrt(a + b*x)*(35*a**2*d**2 - 18*a*b*c*d - 9*b
**2*c**2)/(12*a**2*c**3*(c + d*x)**(3/2)*(a*d - b*c)) + d*sqrt(a + b*x)*(105*a**
3*d**3 - 145*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 9*b**3*c**3)/(12*a**2*c**4*sqrt(
c + d*x)*(a*d - b*c)**2) - (35*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(
c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(4*a**(5/2)*c**(9/2))

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Mathematica [A]  time = 1.25169, size = 221, normalized size = 0.8 \[ \frac{2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{33 a d+9 b c}{a^2 x}+\frac{8 d^3 (9 a d-11 b c)}{(c+d x) (b c-a d)^2}-\frac{8 c d^3}{(c+d x)^2 (b c-a d)}-\frac{6 c}{a x^2}\right )+\frac{3 \log (x) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{a^{5/2}}-\frac{3 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2}}}{24 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*((-6*c)/(a*x^2) + (9*b*c + 33*a*d)/(a^2*x
) - (8*c*d^3)/((b*c - a*d)*(c + d*x)^2) + (8*d^3*(-11*b*c + 9*a*d))/((b*c - a*d)
^2*(c + d*x))) + (3*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*Log[x])/a^(5/2) - (3*(
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^(5/2))/(24*c^(9/2))

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Maple [B]  time = 0.059, size = 1288, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(b*x+a)^(1/2)/a^2/c^4*(108*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^2*c^3*d^3+290*x^3*a^2*b*c*d^4*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)-30*x^3*a*b^2*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+396*x^2*a^
2*b*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-48*x^2*a*b^2*c^3*d^2*(a*c)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)+66*x*a^2*b*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-
6*x*a*b^2*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+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^4*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^4*c^6-210*x^3*a^3*d^5*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)-18*x*b^3*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^3*c^
3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a*b^2*c^5*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+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^4
*b^4*c^4*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^4*c*d^5+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^4*c^5*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^4*c^2*d^4+24*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^3*c^4*d^2-180*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*c^3*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^2*a^2*b^2*c^4*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*b^3*c^5*d-18*x^3*b^3*c^3*d^2*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)-280*x^2*a^3*c*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-36*x^2*b^3
*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*x*a^3*c^2*d^3*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)-24*a^2*b*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-180*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*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^4*a^2*b^2*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^3*c^3*d
^3-360*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
*c^2*d^4)/x^2/(a*c)^(1/2)/(a*d-b*c)^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. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.838024, 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(b*x + a)*(d*x + c)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[-1/48*(4*(6*a*b^2*c^5 - 12*a^2*b*c^4*d + 6*a^3*c^3*d^2 - (9*b^3*c^3*d^2 + 15*a*
b^2*c^2*d^3 - 145*a^2*b*c*d^4 + 105*a^3*d^5)*x^3 - 2*(9*b^3*c^4*d + 12*a*b^2*c^3
*d^2 - 99*a^2*b*c^2*d^3 + 70*a^3*c*d^4)*x^2 - 3*(3*b^3*c^5 + a*b^2*c^4*d - 11*a^
2*b*c^3*d^2 + 7*a^3*c^2*d^3)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*((3*b^
4*c^4*d^2 + 4*a*b^3*c^3*d^3 + 18*a^2*b^2*c^2*d^4 - 60*a^3*b*c*d^5 + 35*a^4*d^6)*
x^4 + 2*(3*b^4*c^5*d + 4*a*b^3*c^4*d^2 + 18*a^2*b^2*c^3*d^3 - 60*a^3*b*c^2*d^4 +
35*a^4*c*d^5)*x^3 + (3*b^4*c^6 + 4*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 - 60*a^3*b*
c^3*d^3 + 35*a^4*c^2*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^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4
*c^4*d^4)*x^4 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^3 + (a^2*b^2
*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^2)*sqrt(a*c)), -1/24*(2*(6*a*b^2*c^5 - 12*
a^2*b*c^4*d + 6*a^3*c^3*d^2 - (9*b^3*c^3*d^2 + 15*a*b^2*c^2*d^3 - 145*a^2*b*c*d^
4 + 105*a^3*d^5)*x^3 - 2*(9*b^3*c^4*d + 12*a*b^2*c^3*d^2 - 99*a^2*b*c^2*d^3 + 70
*a^3*c*d^4)*x^2 - 3*(3*b^3*c^5 + a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 7*a^3*c^2*d^3)
*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 3*((3*b^4*c^4*d^2 + 4*a*b^3*c^3*d^3
+ 18*a^2*b^2*c^2*d^4 - 60*a^3*b*c*d^5 + 35*a^4*d^6)*x^4 + 2*(3*b^4*c^5*d + 4*a*
b^3*c^4*d^2 + 18*a^2*b^2*c^3*d^3 - 60*a^3*b*c^2*d^4 + 35*a^4*c*d^5)*x^3 + (3*b^4
*c^6 + 4*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 - 60*a^3*b*c^3*d^3 + 35*a^4*c^2*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^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^4 + 2*(a^2*b^2*c^7*d
- 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^3 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d
^2)*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(1/x**3/(d*x+c)**(5/2)/(b*x+a)**(1/2),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/(sqrt(b*x + a)*(d*x + c)^(5/2)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError