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

Optimal. Leaf size=128 \[ -\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]

[Out]

(-2*(a + b*x)^(5/2))/(3*d*(c + d*x)^(3/2)) - (10*b*(a + b*x)^(3/2))/(3*d^2*Sqrt[
c + d*x]) + (5*b^2*Sqrt[a + b*x]*Sqrt[c + d*x])/d^3 - (5*b^(3/2)*(b*c - a*d)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/d^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.159719, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x)^(5/2))/(3*d*(c + d*x)^(3/2)) - (10*b*(a + b*x)^(3/2))/(3*d^2*Sqrt[
c + d*x]) + (5*b^2*Sqrt[a + b*x]*Sqrt[c + d*x])/d^3 - (5*b^(3/2)*(b*c - a*d)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/d^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.2381, size = 119, normalized size = 0.93 \[ \frac{5 b^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{7}{2}}} + \frac{5 b^{2} \sqrt{a + b x} \sqrt{c + d x}}{d^{3}} - \frac{10 b \left (a + b x\right )^{\frac{3}{2}}}{3 d^{2} \sqrt{c + d x}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b**(3/2)*(a*d - b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/d**(
7/2) + 5*b**2*sqrt(a + b*x)*sqrt(c + d*x)/d**3 - 10*b*(a + b*x)**(3/2)/(3*d**2*s
qrt(c + d*x)) - 2*(a + b*x)**(5/2)/(3*d*(c + d*x)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.177387, size = 136, normalized size = 1.06 \[ \frac{\sqrt{a+b x} \left (-2 a^2 d^2-2 a b d (5 c+7 d x)+b^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 d^3 (c+d x)^{3/2}}-\frac{5 b^{3/2} (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(-2*a^2*d^2 - 2*a*b*d*(5*c + 7*d*x) + b^2*(15*c^2 + 20*c*d*x + 3*
d^2*x^2)))/(3*d^3*(c + d*x)^(3/2)) - (5*b^(3/2)*(b*c - a*d)*Log[b*c + a*d + 2*b*
d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*d^(7/2))

_______________________________________________________________________________________

Maple [F]  time = 0., size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^(5/2)/(d*x+c)^(5/2),x)

_______________________________________________________________________________________

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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.50274, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, -\frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(15*(b^2*c^3 - a*b*c^2*d + (b^2*c*d^2 - a*b*d^3)*x^2 + 2*(b^2*c^2*d - a*b
*c*d^2)*x)*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*
d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*
d^2)*x) - 4*(3*b^2*d^2*x^2 + 15*b^2*c^2 - 10*a*b*c*d - 2*a^2*d^2 + 2*(10*b^2*c*d
- 7*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3), -
1/6*(15*(b^2*c^3 - a*b*c^2*d + (b^2*c*d^2 - a*b*d^3)*x^2 + 2*(b^2*c^2*d - a*b*c*
d^2)*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)
*d*sqrt(-b/d))) - 2*(3*b^2*d^2*x^2 + 15*b^2*c^2 - 10*a*b*c*d - 2*a^2*d^2 + 2*(10
*b^2*c*d - 7*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2
*d^3)]

_______________________________________________________________________________________

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)/(d*x+c)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.253152, size = 373, normalized size = 2.91 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c d^{4} - a b^{5} d^{5}\right )}{\left (b x + a\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}} + \frac{20 \,{\left (b^{7} c^{2} d^{3} - 2 \, a b^{6} c d^{4} + a^{2} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} + \frac{15 \,{\left (b^{8} c^{3} d^{2} - 3 \, a b^{7} c^{2} d^{3} + 3 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{5 \,{\left (b^{4} c - a b^{3} d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x + a)*(3*(b^6*c*d^4 - a*b^5*d^5)*(b*x + a)/(b^2*c*d^5*abs(b) - a*b*d^6*
abs(b)) + 20*(b^7*c^2*d^3 - 2*a*b^6*c*d^4 + a^2*b^5*d^5)/(b^2*c*d^5*abs(b) - a*b
*d^6*abs(b))) + 15*(b^8*c^3*d^2 - 3*a*b^7*c^2*d^3 + 3*a^2*b^6*c*d^4 - a^3*b^5*d^
5)/(b^2*c*d^5*abs(b) - a*b*d^6*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a
*b*d)^(3/2) + 5*(b^4*c - a*b^3*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c +
(b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^3*abs(b))