3.1510 \(\int \frac{1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=171 \[ \frac{256 d^4 \sqrt{a+b x}}{35 \sqrt{c+d x} (b c-a d)^5}+\frac{128 d^3}{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}+\frac{16 d}{35 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{7 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)} \]

[Out]

-2/(7*(b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x]) + (16*d)/(35*(b*c - a*d)^2*(a +
b*x)^(5/2)*Sqrt[c + d*x]) - (32*d^2)/(35*(b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt[c +
d*x]) + (128*d^3)/(35*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x]) + (256*d^4*Sqr
t[a + b*x])/(35*(b*c - a*d)^5*Sqrt[c + d*x])

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Rubi [A]  time = 0.152757, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{256 d^4 \sqrt{a+b x}}{35 \sqrt{c+d x} (b c-a d)^5}+\frac{128 d^3}{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}+\frac{16 d}{35 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{7 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(9/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(7*(b*c - a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x]) + (16*d)/(35*(b*c - a*d)^2*(a +
b*x)^(5/2)*Sqrt[c + d*x]) - (32*d^2)/(35*(b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt[c +
d*x]) + (128*d^3)/(35*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x]) + (256*d^4*Sqr
t[a + b*x])/(35*(b*c - a*d)^5*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 32.6892, size = 155, normalized size = 0.91 \[ - \frac{256 b d^{3} \sqrt{c + d x}}{35 \sqrt{a + b x} \left (a d - b c\right )^{5}} - \frac{128 d^{3}}{35 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4}} + \frac{32 d^{2}}{35 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{16 d}{35 \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{2}{7 \left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(9/2)/(d*x+c)**(3/2),x)

[Out]

-256*b*d**3*sqrt(c + d*x)/(35*sqrt(a + b*x)*(a*d - b*c)**5) - 128*d**3/(35*sqrt(
a + b*x)*sqrt(c + d*x)*(a*d - b*c)**4) + 32*d**2/(35*(a + b*x)**(3/2)*sqrt(c + d
*x)*(a*d - b*c)**3) + 16*d/(35*(a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)**2) +
2/(7*(a + b*x)**(7/2)*sqrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.387988, size = 120, normalized size = 0.7 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{29 b d^2 (b c-a d)}{(a+b x)^2}+\frac{13 b d (b c-a d)^2}{(a+b x)^3}-\frac{5 b (b c-a d)^3}{(a+b x)^4}+\frac{93 b d^3}{a+b x}+\frac{35 d^4}{c+d x}\right )}{35 (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(9/2)*(c + d*x)^(3/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*((-5*b*(b*c - a*d)^3)/(a + b*x)^4 + (13*b*d*(b*c
- a*d)^2)/(a + b*x)^3 - (29*b*d^2*(b*c - a*d))/(a + b*x)^2 + (93*b*d^3)/(a + b*x
) + (35*d^4)/(c + d*x)))/(35*(b*c - a*d)^5)

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Maple [A]  time = 0.015, size = 256, normalized size = 1.5 \[ -{\frac{256\,{b}^{4}{d}^{4}{x}^{4}+896\,a{b}^{3}{d}^{4}{x}^{3}+128\,{b}^{4}c{d}^{3}{x}^{3}+1120\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}+448\,a{b}^{3}c{d}^{3}{x}^{2}-32\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+560\,{a}^{3}b{d}^{4}x+560\,{a}^{2}{b}^{2}c{d}^{3}x-112\,a{b}^{3}{c}^{2}{d}^{2}x+16\,{b}^{4}{c}^{3}dx+70\,{a}^{4}{d}^{4}+280\,{a}^{3}bc{d}^{3}-140\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}+56\,a{b}^{3}{c}^{3}d-10\,{b}^{4}{c}^{4}}{35\,{a}^{5}{d}^{5}-175\,{a}^{4}bc{d}^{4}+350\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-350\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+175\,a{b}^{4}{c}^{4}d-35\,{b}^{5}{c}^{5}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(9/2)/(d*x+c)^(3/2),x)

[Out]

-2/35*(128*b^4*d^4*x^4+448*a*b^3*d^4*x^3+64*b^4*c*d^3*x^3+560*a^2*b^2*d^4*x^2+22
4*a*b^3*c*d^3*x^2-16*b^4*c^2*d^2*x^2+280*a^3*b*d^4*x+280*a^2*b^2*c*d^3*x-56*a*b^
3*c^2*d^2*x+8*b^4*c^3*d*x+35*a^4*d^4+140*a^3*b*c*d^3-70*a^2*b^2*c^2*d^2+28*a*b^3
*c^3*d-5*b^4*c^4)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*
c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)

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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)^(9/2)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89178, size = 930, normalized size = 5.44 \[ \frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} - 5 \, b^{4} c^{4} + 28 \, a b^{3} c^{3} d - 70 \, a^{2} b^{2} c^{2} d^{2} + 140 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4} + 64 \,{\left (b^{4} c d^{3} + 7 \, a b^{3} d^{4}\right )} x^{3} - 16 \,{\left (b^{4} c^{2} d^{2} - 14 \, a b^{3} c d^{3} - 35 \, a^{2} b^{2} d^{4}\right )} x^{2} + 8 \,{\left (b^{4} c^{3} d - 7 \, a b^{3} c^{2} d^{2} + 35 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{35 \,{\left (a^{4} b^{5} c^{6} - 5 \, a^{5} b^{4} c^{5} d + 10 \, a^{6} b^{3} c^{4} d^{2} - 10 \, a^{7} b^{2} c^{3} d^{3} + 5 \, a^{8} b c^{2} d^{4} - a^{9} c d^{5} +{\left (b^{9} c^{5} d - 5 \, a b^{8} c^{4} d^{2} + 10 \, a^{2} b^{7} c^{3} d^{3} - 10 \, a^{3} b^{6} c^{2} d^{4} + 5 \, a^{4} b^{5} c d^{5} - a^{5} b^{4} d^{6}\right )} x^{5} +{\left (b^{9} c^{6} - a b^{8} c^{5} d - 10 \, a^{2} b^{7} c^{4} d^{2} + 30 \, a^{3} b^{6} c^{3} d^{3} - 35 \, a^{4} b^{5} c^{2} d^{4} + 19 \, a^{5} b^{4} c d^{5} - 4 \, a^{6} b^{3} d^{6}\right )} x^{4} + 2 \,{\left (2 \, a b^{8} c^{6} - 7 \, a^{2} b^{7} c^{5} d + 5 \, a^{3} b^{6} c^{4} d^{2} + 10 \, a^{4} b^{5} c^{3} d^{3} - 20 \, a^{5} b^{4} c^{2} d^{4} + 13 \, a^{6} b^{3} c d^{5} - 3 \, a^{7} b^{2} d^{6}\right )} x^{3} + 2 \,{\left (3 \, a^{2} b^{7} c^{6} - 13 \, a^{3} b^{6} c^{5} d + 20 \, a^{4} b^{5} c^{4} d^{2} - 10 \, a^{5} b^{4} c^{3} d^{3} - 5 \, a^{6} b^{3} c^{2} d^{4} + 7 \, a^{7} b^{2} c d^{5} - 2 \, a^{8} b d^{6}\right )} x^{2} +{\left (4 \, a^{3} b^{6} c^{6} - 19 \, a^{4} b^{5} c^{5} d + 35 \, a^{5} b^{4} c^{4} d^{2} - 30 \, a^{6} b^{3} c^{3} d^{3} + 10 \, a^{7} b^{2} c^{2} d^{4} + a^{8} b c d^{5} - a^{9} d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/2)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

2/35*(128*b^4*d^4*x^4 - 5*b^4*c^4 + 28*a*b^3*c^3*d - 70*a^2*b^2*c^2*d^2 + 140*a^
3*b*c*d^3 + 35*a^4*d^4 + 64*(b^4*c*d^3 + 7*a*b^3*d^4)*x^3 - 16*(b^4*c^2*d^2 - 14
*a*b^3*c*d^3 - 35*a^2*b^2*d^4)*x^2 + 8*(b^4*c^3*d - 7*a*b^3*c^2*d^2 + 35*a^2*b^2
*c*d^3 + 35*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b^5*c^6 - 5*a^5*b^4*c
^5*d + 10*a^6*b^3*c^4*d^2 - 10*a^7*b^2*c^3*d^3 + 5*a^8*b*c^2*d^4 - a^9*c*d^5 + (
b^9*c^5*d - 5*a*b^8*c^4*d^2 + 10*a^2*b^7*c^3*d^3 - 10*a^3*b^6*c^2*d^4 + 5*a^4*b^
5*c*d^5 - a^5*b^4*d^6)*x^5 + (b^9*c^6 - a*b^8*c^5*d - 10*a^2*b^7*c^4*d^2 + 30*a^
3*b^6*c^3*d^3 - 35*a^4*b^5*c^2*d^4 + 19*a^5*b^4*c*d^5 - 4*a^6*b^3*d^6)*x^4 + 2*(
2*a*b^8*c^6 - 7*a^2*b^7*c^5*d + 5*a^3*b^6*c^4*d^2 + 10*a^4*b^5*c^3*d^3 - 20*a^5*
b^4*c^2*d^4 + 13*a^6*b^3*c*d^5 - 3*a^7*b^2*d^6)*x^3 + 2*(3*a^2*b^7*c^6 - 13*a^3*
b^6*c^5*d + 20*a^4*b^5*c^4*d^2 - 10*a^5*b^4*c^3*d^3 - 5*a^6*b^3*c^2*d^4 + 7*a^7*
b^2*c*d^5 - 2*a^8*b*d^6)*x^2 + (4*a^3*b^6*c^6 - 19*a^4*b^5*c^5*d + 35*a^5*b^4*c^
4*d^2 - 30*a^6*b^3*c^3*d^3 + 10*a^7*b^2*c^2*d^4 + a^8*b*c*d^5 - a^9*d^6)*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)**(9/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.710606, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/2)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

Done