3.1509 \(\int \frac{1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{32 d^3 \sqrt{a+b x}}{5 \sqrt{c+d x} (b c-a d)^4}-\frac{16 d^2}{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}+\frac{4 d}{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)} \]

[Out]

-2/(5*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (4*d)/(5*(b*c - a*d)^2*(a + b
*x)^(3/2)*Sqrt[c + d*x]) - (16*d^2)/(5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x]
) - (32*d^3*Sqrt[a + b*x])/(5*(b*c - a*d)^4*Sqrt[c + d*x])

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Rubi [A]  time = 0.114606, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{32 d^3 \sqrt{a+b x}}{5 \sqrt{c+d x} (b c-a d)^4}-\frac{16 d^2}{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}+\frac{4 d}{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(7/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(5*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (4*d)/(5*(b*c - a*d)^2*(a + b
*x)^(3/2)*Sqrt[c + d*x]) - (16*d^2)/(5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x]
) - (32*d^3*Sqrt[a + b*x])/(5*(b*c - a*d)^4*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 22.7227, size = 121, normalized size = 0.89 \[ - \frac{32 d^{3} \sqrt{a + b x}}{5 \sqrt{c + d x} \left (a d - b c\right )^{4}} + \frac{16 d^{2}}{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{4 d}{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{2}{5 \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(7/2)/(d*x+c)**(3/2),x)

[Out]

-32*d**3*sqrt(a + b*x)/(5*sqrt(c + d*x)*(a*d - b*c)**4) + 16*d**2/(5*sqrt(a + b*
x)*sqrt(c + d*x)*(a*d - b*c)**3) + 4*d/(5*(a + b*x)**(3/2)*sqrt(c + d*x)*(a*d -
b*c)**2) + 2/(5*(a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.165824, size = 112, normalized size = 0.82 \[ \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 d^3}{(c+d x) (b c-a d)^4}-\frac{22 b d^2}{5 (a+b x) (b c-a d)^4}+\frac{6 b d}{5 (a+b x)^2 (b c-a d)^3}-\frac{2 b}{5 (a+b x)^3 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(7/2)*(c + d*x)^(3/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*((-2*b)/(5*(b*c - a*d)^2*(a + b*x)^3) + (6*b*d)/(5*(
b*c - a*d)^3*(a + b*x)^2) - (22*b*d^2)/(5*(b*c - a*d)^4*(a + b*x)) - (2*d^3)/((b
*c - a*d)^4*(c + d*x)))

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Maple [A]  time = 0.013, size = 170, normalized size = 1.3 \[ -{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+80\,a{b}^{2}{d}^{3}{x}^{2}+16\,{b}^{3}c{d}^{2}{x}^{2}+60\,{a}^{2}b{d}^{3}x+40\,a{b}^{2}c{d}^{2}x-4\,{b}^{3}{c}^{2}dx+10\,{a}^{3}{d}^{3}+30\,{a}^{2}bc{d}^{2}-10\,a{b}^{2}{c}^{2}d+2\,{b}^{3}{c}^{3}}{5\,{d}^{4}{a}^{4}-20\,b{d}^{3}c{a}^{3}+30\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-20\,{b}^{3}d{c}^{3}a+5\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(7/2)/(d*x+c)^(3/2),x)

[Out]

-2/5*(16*b^3*d^3*x^3+40*a*b^2*d^3*x^2+8*b^3*c*d^2*x^2+30*a^2*b*d^3*x+20*a*b^2*c*
d^2*x-2*b^3*c^2*d*x+5*a^3*d^3+15*a^2*b*c*d^2-5*a*b^2*c^2*d+b^3*c^3)/(b*x+a)^(5/2
)/(d*x+c)^(1/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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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(1/((b*x + a)^(7/2)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.788255, size = 614, normalized size = 4.51 \[ -\frac{2 \,{\left (16 \, b^{3} d^{3} x^{3} + b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 8 \,{\left (b^{3} c d^{2} + 5 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (b^{3} c^{2} d - 10 \, a b^{2} c d^{2} - 15 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{5 \,{\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} +{\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} +{\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \,{\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} +{\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/2)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

-2/5*(16*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 5*a^3*d^3 + 8*
(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 10*a*b^2*c*d^2 - 15*a^2*b*d^3)*x)
*sqrt(b*x + a)*sqrt(d*x + c)/(a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2
- 4*a^6*b*c^2*d^3 + a^7*c*d^4 + (b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3
- 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d
^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*x^3 + 3*(a*b^6*c^5 -
3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6
*b*d^5)*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2
*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*x)

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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/(b*x+a)**(7/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.437952, size = 1121, normalized size = 8.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/2)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

-2*sqrt(b*x + a)*b^2*d^3/((b^4*c^4*abs(b) - 4*a*b^3*c^3*d*abs(b) + 6*a^2*b^2*c^2
*d^2*abs(b) - 4*a^3*b*c*d^3*abs(b) + a^4*d^4*abs(b))*sqrt(b^2*c + (b*x + a)*b*d
- a*b*d)) - 4/5*(11*sqrt(b*d)*b^10*c^4*d^2 - 44*sqrt(b*d)*a*b^9*c^3*d^3 + 66*sqr
t(b*d)*a^2*b^8*c^2*d^4 - 44*sqrt(b*d)*a^3*b^7*c*d^5 + 11*sqrt(b*d)*a^4*b^6*d^6 -
50*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*
b^8*c^3*d^2 + 150*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^2*a*b^7*c^2*d^3 - 150*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^2*a^2*b^6*c*d^4 + 50*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*d^5 + 80*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^2*d^2 - 160*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^5*c*
d^3 + 80*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^4*a^2*b^4*d^4 - 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a
)*b*d - a*b*d))^6*b^4*c*d^2 + 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^6*a*b^3*d^3 + 5*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^2*d^2)/((b^3*c^3*abs(b) - 3*a*b^2*c^2*d
*abs(b) + 3*a^2*b*c*d^2*abs(b) - a^3*d^3*abs(b))*(b^2*c - a*b*d - (sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^5)