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

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

[Out]

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

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Rubi [A]  time = 0.16046, 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 \[ \frac{5 d^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2}}+\frac{5 d^2 \sqrt{a+b x} \sqrt{c+d x}}{b^3}-\frac{10 d (c+d x)^{3/2}}{3 b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.3511, size = 119, normalized size = 0.93 \[ - \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} - \frac{10 d \left (c + d x\right )^{\frac{3}{2}}}{3 b^{2} \sqrt{a + b x}} + \frac{5 d^{2} \sqrt{a + b x} \sqrt{c + d x}}{b^{3}} - \frac{5 d^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c + d*x)**(5/2)/(3*b*(a + b*x)**(3/2)) - 10*d*(c + d*x)**(3/2)/(3*b**2*sqrt(
a + b*x)) + 5*d**2*sqrt(a + b*x)*sqrt(c + d*x)/b**3 - 5*d**(3/2)*(a*d - b*c)*ata
nh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/b**(7/2)

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Mathematica [A]  time = 0.173409, size = 134, normalized size = 1.05 \[ \frac{\sqrt{c+d x} \left (15 a^2 d^2-10 a b d (c-2 d x)+b^2 \left (-2 c^2-14 c d x+3 d^2 x^2\right )\right )}{3 b^3 (a+b x)^{3/2}}+\frac{5 d^{3/2} (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(15*a^2*d^2 - 10*a*b*d*(c - 2*d*x) + b^2*(-2*c^2 - 14*c*d*x + 3*d
^2*x^2)))/(3*b^3*(a + b*x)^(3/2)) + (5*d^(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*b^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.491748, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a^{2} b c d - a^{3} d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (3 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 2 \,{\left (7 \, b^{2} c d - 10 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, \frac{15 \,{\left (a^{2} b c d - a^{3} d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) + 2 \,{\left (3 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 2 \,{\left (7 \, b^{2} c d - 10 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.654344, 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)^(5/2),x, algorithm="giac")

[Out]

sage0*x