3.1520 \(\int \frac{1}{(a+b x)^{7/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=172 \[ -\frac{256 b d^3 \sqrt{a+b x}}{15 \sqrt{c+d x} (b c-a d)^5}-\frac{128 d^3 \sqrt{a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac{16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

-2/(5*(b*c - a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2)) + (16*d)/(15*(b*c - a*d)^2*(a
+ b*x)^(3/2)*(c + d*x)^(3/2)) - (32*d^2)/(5*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*
x)^(3/2)) - (128*d^3*Sqrt[a + b*x])/(15*(b*c - a*d)^4*(c + d*x)^(3/2)) - (256*b*
d^3*Sqrt[a + b*x])/(15*(b*c - a*d)^5*Sqrt[c + d*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.159121, antiderivative size = 172, 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 \[ -\frac{256 b d^3 \sqrt{a+b x}}{15 \sqrt{c+d x} (b c-a d)^5}-\frac{128 d^3 \sqrt{a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac{16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(7/2)*(c + d*x)^(5/2)),x]

[Out]

-2/(5*(b*c - a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2)) + (16*d)/(15*(b*c - a*d)^2*(a
+ b*x)^(3/2)*(c + d*x)^(3/2)) - (32*d^2)/(5*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*
x)^(3/2)) - (128*d^3*Sqrt[a + b*x])/(15*(b*c - a*d)^4*(c + d*x)^(3/2)) - (256*b*
d^3*Sqrt[a + b*x])/(15*(b*c - a*d)^5*Sqrt[c + d*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.7655, size = 155, normalized size = 0.9 \[ \frac{256 b d^{3} \sqrt{a + b x}}{15 \sqrt{c + d x} \left (a d - b c\right )^{5}} - \frac{128 d^{3} \sqrt{a + b x}}{15 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{32 d^{2}}{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{16 d}{15 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{2}{5 \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(7/2)/(d*x+c)**(5/2),x)

[Out]

256*b*d**3*sqrt(a + b*x)/(15*sqrt(c + d*x)*(a*d - b*c)**5) - 128*d**3*sqrt(a + b
*x)/(15*(c + d*x)**(3/2)*(a*d - b*c)**4) + 32*d**2/(5*sqrt(a + b*x)*(c + d*x)**(
3/2)*(a*d - b*c)**3) + 16*d/(15*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)**2
) + 2/(5*(a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.385134, size = 124, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{14 b^2 d (b c-a d)}{(a+b x)^2}-\frac{3 b^2 (b c-a d)^2}{(a+b x)^3}-\frac{73 b^2 d^2}{a+b x}+\frac{5 d^3 (a d-b c)}{(c+d x)^2}-\frac{55 b d^3}{c+d x}\right )}{15 (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(7/2)*(c + d*x)^(5/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*((-3*b^2*(b*c - a*d)^2)/(a + b*x)^3 + (14*b^2*d*(
b*c - a*d))/(a + b*x)^2 - (73*b^2*d^2)/(a + b*x) + (5*d^3*(-(b*c) + a*d))/(c + d
*x)^2 - (55*b*d^3)/(c + d*x)))/(15*(b*c - a*d)^5)

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 256, normalized size = 1.5 \[ -{\frac{-256\,{b}^{4}{d}^{4}{x}^{4}-640\,a{b}^{3}{d}^{4}{x}^{3}-384\,{b}^{4}c{d}^{3}{x}^{3}-480\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-960\,a{b}^{3}c{d}^{3}{x}^{2}-96\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}-80\,{a}^{3}b{d}^{4}x-720\,{a}^{2}{b}^{2}c{d}^{3}x-240\,a{b}^{3}{c}^{2}{d}^{2}x+16\,{b}^{4}{c}^{3}dx+10\,{a}^{4}{d}^{4}-120\,{a}^{3}bc{d}^{3}-180\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}+40\,a{b}^{3}{c}^{3}d-6\,{b}^{4}{c}^{4}}{15\,{a}^{5}{d}^{5}-75\,{a}^{4}bc{d}^{4}+150\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-150\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+75\,a{b}^{4}{c}^{4}d-15\,{b}^{5}{c}^{5}} \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(7/2)/(d*x+c)^(5/2),x)

[Out]

-2/15*(-128*b^4*d^4*x^4-320*a*b^3*d^4*x^3-192*b^4*c*d^3*x^3-240*a^2*b^2*d^4*x^2-
480*a*b^3*c*d^3*x^2-48*b^4*c^2*d^2*x^2-40*a^3*b*d^4*x-360*a^2*b^2*c*d^3*x-120*a*
b^3*c^2*d^2*x+8*b^4*c^3*d*x+5*a^4*d^4-60*a^3*b*c*d^3-90*a^2*b^2*c^2*d^2+20*a*b^3
*c^3*d-3*b^4*c^4)/(b*x+a)^(5/2)/(d*x+c)^(3/2)/(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)

_______________________________________________________________________________________

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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.09969, size = 965, normalized size = 5.61 \[ -\frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} + 3 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 90 \, a^{2} b^{2} c^{2} d^{2} + 60 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4} + 64 \,{\left (3 \, b^{4} c d^{3} + 5 \, a b^{3} d^{4}\right )} x^{3} + 48 \,{\left (b^{4} c^{2} d^{2} + 10 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \,{\left (b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} - 45 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{15 \,{\left (a^{3} b^{5} c^{7} - 5 \, a^{4} b^{4} c^{6} d + 10 \, a^{5} b^{3} c^{5} d^{2} - 10 \, a^{6} b^{2} c^{4} d^{3} + 5 \, a^{7} b c^{3} d^{4} - a^{8} c^{2} d^{5} +{\left (b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} + 10 \, a^{2} b^{6} c^{3} d^{4} - 10 \, a^{3} b^{5} c^{2} d^{5} + 5 \, a^{4} b^{4} c d^{6} - a^{5} b^{3} d^{7}\right )} x^{5} +{\left (2 \, b^{8} c^{6} d - 7 \, a b^{7} c^{5} d^{2} + 5 \, a^{2} b^{6} c^{4} d^{3} + 10 \, a^{3} b^{5} c^{3} d^{4} - 20 \, a^{4} b^{4} c^{2} d^{5} + 13 \, a^{5} b^{3} c d^{6} - 3 \, a^{6} b^{2} d^{7}\right )} x^{4} +{\left (b^{8} c^{7} + a b^{7} c^{6} d - 17 \, a^{2} b^{6} c^{5} d^{2} + 35 \, a^{3} b^{5} c^{4} d^{3} - 25 \, a^{4} b^{4} c^{3} d^{4} - a^{5} b^{3} c^{2} d^{5} + 9 \, a^{6} b^{2} c d^{6} - 3 \, a^{7} b d^{7}\right )} x^{3} +{\left (3 \, a b^{7} c^{7} - 9 \, a^{2} b^{6} c^{6} d + a^{3} b^{5} c^{5} d^{2} + 25 \, a^{4} b^{4} c^{4} d^{3} - 35 \, a^{5} b^{3} c^{3} d^{4} + 17 \, a^{6} b^{2} c^{2} d^{5} - a^{7} b c d^{6} - a^{8} d^{7}\right )} x^{2} +{\left (3 \, a^{2} b^{6} c^{7} - 13 \, a^{3} b^{5} c^{6} d + 20 \, a^{4} b^{4} c^{5} d^{2} - 10 \, a^{5} b^{3} c^{4} d^{3} - 5 \, a^{6} b^{2} c^{3} d^{4} + 7 \, a^{7} b c^{2} d^{5} - 2 \, a^{8} c d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/2)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

-2/15*(128*b^4*d^4*x^4 + 3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^
3*b*c*d^3 - 5*a^4*d^4 + 64*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 48*(b^4*c^2*d^2 + 1
0*a*b^3*c*d^3 + 5*a^2*b^2*d^4)*x^2 - 8*(b^4*c^3*d - 15*a*b^3*c^2*d^2 - 45*a^2*b^
2*c*d^3 - 5*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^3*b^5*c^7 - 5*a^4*b^4*c
^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5 +
(b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^3*b^5*c^2*d^5 + 5*a^
4*b^4*c*d^6 - a^5*b^3*d^7)*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*
d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7
)*x^4 + (b^8*c^7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^
4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*a^7*b*d^7)*x^3 + (3*a*b^7*
c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^3*d^
4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^
5*c^6*d + 20*a^4*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*
c^2*d^5 - 2*a^8*c*d^6)*x)

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.826491, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/2)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

Done