3.1487 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx\)

Optimal. Leaf size=120 \[ \frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2}}-\frac{2 d^2 \sqrt{c+d x}}{b^3 \sqrt{a+b x}}-\frac{2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}} \]

[Out]

(-2*d^2*Sqrt[c + d*x])/(b^3*Sqrt[a + b*x]) - (2*d*(c + d*x)^(3/2))/(3*b^2*(a + b
*x)^(3/2)) - (2*(c + d*x)^(5/2))/(5*b*(a + b*x)^(5/2)) + (2*d^(5/2)*ArcTanh[(Sqr
t[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/b^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.137416, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2}}-\frac{2 d^2 \sqrt{c+d x}}{b^3 \sqrt{a+b x}}-\frac{2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(a + b*x)^(7/2),x]

[Out]

(-2*d^2*Sqrt[c + d*x])/(b^3*Sqrt[a + b*x]) - (2*d*(c + d*x)^(3/2))/(3*b^2*(a + b
*x)^(3/2)) - (2*(c + d*x)^(5/2))/(5*b*(a + b*x)^(5/2)) + (2*d^(5/2)*ArcTanh[(Sqr
t[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/b^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.7362, size = 112, normalized size = 0.93 \[ - \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 b \left (a + b x\right )^{\frac{5}{2}}} - \frac{2 d \left (c + d x\right )^{\frac{3}{2}}}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 d^{2} \sqrt{c + d x}}{b^{3} \sqrt{a + b x}} + \frac{2 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/(b*x+a)**(7/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.229234, size = 121, normalized size = 1.01 \[ \frac{d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{7/2}}-\frac{2 \sqrt{c+d x} \left (11 d (a+b x) (b c-a d)+3 (b c-a d)^2+23 d^2 (a+b x)^2\right )}{15 b^3 (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(a + b*x)^(7/2),x]

[Out]

(-2*Sqrt[c + d*x]*(3*(b*c - a*d)^2 + 11*d*(b*c - a*d)*(a + b*x) + 23*d^2*(a + b*
x)^2))/(15*b^3*(a + b*x)^(5/2)) + (d^(5/2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*S
qrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/b^(7/2)

_______________________________________________________________________________________

Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^(7/2),x)

[Out]

int((d*x+c)^(5/2)/(b*x+a)^(7/2),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((d*x + c)^(5/2)/(b*x + a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.595071, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{30 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, \frac{15 \,{\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) - 2 \,{\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{15 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(7/2),x, algorithm="fricas")

[Out]

[1/30*(15*(b^3*d^2*x^3 + 3*a*b^2*d^2*x^2 + 3*a^2*b*d^2*x + a^3*d^2)*sqrt(d/b)*lo
g(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*
sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(23*b^2*d^2
*x^2 + 3*b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + (11*b^2*c*d + 35*a*b*d^2)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^4*x + a^3*b^3), 1/15*(15*
(b^3*d^2*x^3 + 3*a*b^2*d^2*x^2 + 3*a^2*b*d^2*x + a^3*d^2)*sqrt(-d/b)*arctan(1/2*
(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) - 2*(23*b^2*d^
2*x^2 + 3*b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + (11*b^2*c*d + 35*a*b*d^2)*x)*sqrt(b
*x + a)*sqrt(d*x + c))/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^4*x + a^3*b^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((d*x+c)**(5/2)/(b*x+a)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.639029, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(7/2),x, algorithm="giac")

[Out]

sage0*x