3.676 \(\int \frac{(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=226 \[ -\frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{9/2}}+\frac{5 \sqrt{a+b x} (b c-7 a d) (b c-a d)^2}{8 a c^4 \sqrt{c+d x}}-\frac{5 (a+b x)^{3/2} (b c-7 a d) (b c-a d)}{24 a c^3 x \sqrt{c+d x}}-\frac{(a+b x)^{5/2} (b c-7 a d)}{12 a c^2 x^2 \sqrt{c+d x}}-\frac{(a+b x)^{7/2}}{3 a c x^3 \sqrt{c+d x}} \]

[Out]

(5*(b*c - 7*a*d)*(b*c - a*d)^2*Sqrt[a + b*x])/(8*a*c^4*Sqrt[c + d*x]) - (5*(b*c
- 7*a*d)*(b*c - a*d)*(a + b*x)^(3/2))/(24*a*c^3*x*Sqrt[c + d*x]) - ((b*c - 7*a*d
)*(a + b*x)^(5/2))/(12*a*c^2*x^2*Sqrt[c + d*x]) - (a + b*x)^(7/2)/(3*a*c*x^3*Sqr
t[c + d*x]) - (5*(b*c - 7*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sq
rt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(9/2))

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Rubi [A]  time = 0.408488, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{9/2}}+\frac{5 \sqrt{a+b x} (b c-7 a d) (b c-a d)^2}{8 a c^4 \sqrt{c+d x}}-\frac{5 (a+b x)^{3/2} (b c-7 a d) (b c-a d)}{24 a c^3 x \sqrt{c+d x}}-\frac{(a+b x)^{5/2} (b c-7 a d)}{12 a c^2 x^2 \sqrt{c+d x}}-\frac{(a+b x)^{7/2}}{3 a c x^3 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/(x^4*(c + d*x)^(3/2)),x]

[Out]

(5*(b*c - 7*a*d)*(b*c - a*d)^2*Sqrt[a + b*x])/(8*a*c^4*Sqrt[c + d*x]) - (5*(b*c
- 7*a*d)*(b*c - a*d)*(a + b*x)^(3/2))/(24*a*c^3*x*Sqrt[c + d*x]) - ((b*c - 7*a*d
)*(a + b*x)^(5/2))/(12*a*c^2*x^2*Sqrt[c + d*x]) - (a + b*x)^(7/2)/(3*a*c*x^3*Sqr
t[c + d*x]) - (5*(b*c - 7*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sq
rt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(9/2))

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Rubi in Sympy [A]  time = 38.1522, size = 207, normalized size = 0.92 \[ \frac{2 d \left (a + b x\right )^{\frac{7}{2}}}{c x^{3} \sqrt{c + d x} \left (a d - b c\right )} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (7 a d - b c\right )}{3 c^{2} x^{3} \left (a d - b c\right )} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (7 a d - b c\right )}{12 c^{3} x^{2}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (7 a d - b c\right )}{8 c^{4} x} + \frac{5 \left (a d - b c\right )^{2} \left (7 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 \sqrt{a} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)/x**4/(d*x+c)**(3/2),x)

[Out]

2*d*(a + b*x)**(7/2)/(c*x**3*sqrt(c + d*x)*(a*d - b*c)) - (a + b*x)**(5/2)*sqrt(
c + d*x)*(7*a*d - b*c)/(3*c**2*x**3*(a*d - b*c)) + 5*(a + b*x)**(3/2)*sqrt(c + d
*x)*(7*a*d - b*c)/(12*c**3*x**2) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)*(7*
a*d - b*c)/(8*c**4*x) + 5*(a*d - b*c)**2*(7*a*d - b*c)*atanh(sqrt(c)*sqrt(a + b*
x)/(sqrt(a)*sqrt(c + d*x)))/(8*sqrt(a)*c**(9/2))

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Mathematica [A]  time = 0.341041, size = 214, normalized size = 0.95 \[ \frac{-\frac{2 \sqrt{c} \sqrt{a+b x} \left (a^2 \left (8 c^3-14 c^2 d x+35 c d^2 x^2+105 d^3 x^3\right )+2 a b c x \left (13 c^2-34 c d x-95 d^2 x^2\right )+3 b^2 c^2 x^2 (11 c+27 d x)\right )}{x^3 \sqrt{c+d x}}+\frac{15 \log (x) (b c-7 a d) (b c-a d)^2}{\sqrt{a}}+\frac{15 (7 a d-b c) (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}}{48 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/(x^4*(c + d*x)^(3/2)),x]

[Out]

((-2*Sqrt[c]*Sqrt[a + b*x]*(3*b^2*c^2*x^2*(11*c + 27*d*x) + 2*a*b*c*x*(13*c^2 -
34*c*d*x - 95*d^2*x^2) + a^2*(8*c^3 - 14*c^2*d*x + 35*c*d^2*x^2 + 105*d^3*x^3)))
/(x^3*Sqrt[c + d*x]) + (15*(b*c - 7*a*d)*(b*c - a*d)^2*Log[x])/Sqrt[a] + (15*(b*
c - a*d)^2*(-(b*c) + 7*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]])/Sqrt[a])/(48*c^(9/2))

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Maple [B]  time = 0.045, size = 704, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x^4/(d*x+c)^(3/2),x)

[Out]

1/48*(b*x+a)^(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^3*d^4-225*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^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^4*a*b^2*c^2*d^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*b^3*c^3*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^3*a^3*c*d^3-225*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^2+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^3*a*b^2*c^3*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*b^3*c^4-210*x^3*a^2*d^3*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)+380*x^3*a*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-162*x^3*b^2*
c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-70*x^2*a^2*c*d^2*(a*c)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)+136*x^2*a*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-66*x^2*b^2*
c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+28*x*a^2*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)-52*x*a*b*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-16*a^2*c^3*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2))/c^4/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^3/(d*x+c)^
(1/2)

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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((b*x + a)^(5/2)/((d*x + c)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.799218, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (8 \, a^{2} c^{3} +{\left (81 \, b^{2} c^{2} d - 190 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} +{\left (33 \, b^{2} c^{3} - 68 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{2} + 2 \,{\left (13 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left ({\left (b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 15 \, a^{2} b c^{2} d^{2} - 7 \, a^{3} c d^{3}\right )} x^{3}\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{96 \,{\left (c^{4} d x^{4} + c^{5} x^{3}\right )} \sqrt{a c}}, -\frac{2 \,{\left (8 \, a^{2} c^{3} +{\left (81 \, b^{2} c^{2} d - 190 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} +{\left (33 \, b^{2} c^{3} - 68 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{2} + 2 \,{\left (13 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left ({\left (b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 15 \, a^{2} b c^{2} d^{2} - 7 \, a^{3} c d^{3}\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{48 \,{\left (c^{4} d x^{4} + c^{5} x^{3}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/((d*x + c)^(3/2)*x^4),x, algorithm="fricas")

[Out]

[-1/96*(4*(8*a^2*c^3 + (81*b^2*c^2*d - 190*a*b*c*d^2 + 105*a^2*d^3)*x^3 + (33*b^
2*c^3 - 68*a*b*c^2*d + 35*a^2*c*d^2)*x^2 + 2*(13*a*b*c^3 - 7*a^2*c^2*d)*x)*sqrt(
a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((b^3*c^3*d - 9*a*b^2*c^2*d^2 + 15*a^2*b*c
*d^3 - 7*a^3*d^4)*x^4 + (b^3*c^4 - 9*a*b^2*c^3*d + 15*a^2*b*c^2*d^2 - 7*a^3*c*d^
3)*x^3)*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)*sqr
t(a*c))/x^2))/((c^4*d*x^4 + c^5*x^3)*sqrt(a*c)), -1/48*(2*(8*a^2*c^3 + (81*b^2*c
^2*d - 190*a*b*c*d^2 + 105*a^2*d^3)*x^3 + (33*b^2*c^3 - 68*a*b*c^2*d + 35*a^2*c*
d^2)*x^2 + 2*(13*a*b*c^3 - 7*a^2*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c
) + 15*((b^3*c^3*d - 9*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - 7*a^3*d^4)*x^4 + (b^3*c^
4 - 9*a*b^2*c^3*d + 15*a^2*b*c^2*d^2 - 7*a^3*c*d^3)*x^3)*arctan(1/2*(2*a*c + (b*
c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((c^4*d*x^4 + c^5*x^3
)*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)**(5/2)/x**4/(d*x+c)**(3/2),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((b*x + a)^(5/2)/((d*x + c)^(3/2)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError