3.1521 \(\int \frac{1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=207 \[ \frac{512 b d^4 \sqrt{a+b x}}{21 \sqrt{c+d x} (b c-a d)^6}+\frac{256 d^4 \sqrt{a+b x}}{21 (c+d x)^{3/2} (b c-a d)^5}+\frac{64 d^3}{7 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{21 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^3}+\frac{4 d}{7 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

-2/(7*(b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2)) + (4*d)/(7*(b*c - a*d)^2*(a +
b*x)^(5/2)*(c + d*x)^(3/2)) - (32*d^2)/(21*(b*c - a*d)^3*(a + b*x)^(3/2)*(c + d
*x)^(3/2)) + (64*d^3)/(7*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (256*d^4
*Sqrt[a + b*x])/(21*(b*c - a*d)^5*(c + d*x)^(3/2)) + (512*b*d^4*Sqrt[a + b*x])/(
21*(b*c - a*d)^6*Sqrt[c + d*x])

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Rubi [A]  time = 0.203867, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{512 b d^4 \sqrt{a+b x}}{21 \sqrt{c+d x} (b c-a d)^6}+\frac{256 d^4 \sqrt{a+b x}}{21 (c+d x)^{3/2} (b c-a d)^5}+\frac{64 d^3}{7 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{21 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^3}+\frac{4 d}{7 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(9/2)*(c + d*x)^(5/2)),x]

[Out]

-2/(7*(b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2)) + (4*d)/(7*(b*c - a*d)^2*(a +
b*x)^(5/2)*(c + d*x)^(3/2)) - (32*d^2)/(21*(b*c - a*d)^3*(a + b*x)^(3/2)*(c + d
*x)^(3/2)) + (64*d^3)/(7*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (256*d^4
*Sqrt[a + b*x])/(21*(b*c - a*d)^5*(c + d*x)^(3/2)) + (512*b*d^4*Sqrt[a + b*x])/(
21*(b*c - a*d)^6*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 44.885, size = 190, normalized size = 0.92 \[ \frac{512 b^{2} d^{3} \sqrt{c + d x}}{21 \sqrt{a + b x} \left (a d - b c\right )^{6}} + \frac{256 b d^{3}}{21 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{5}} - \frac{64 d^{3}}{21 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{32 d^{2}}{21 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{4 d}{7 \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{2}{7 \left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(9/2)/(d*x+c)**(5/2),x)

[Out]

512*b**2*d**3*sqrt(c + d*x)/(21*sqrt(a + b*x)*(a*d - b*c)**6) + 256*b*d**3/(21*s
qrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**5) - 64*d**3/(21*sqrt(a + b*x)*(c + d*x)
**(3/2)*(a*d - b*c)**4) + 32*d**2/(21*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b
*c)**3) + 4*d/(7*(a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*d - b*c)**2) + 2/(7*(a + b
*x)**(7/2)*(c + d*x)**(3/2)*(a*d - b*c))

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Mathematica [A]  time = 0.571397, size = 149, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{37 b^2 d^2 (b c-a d)}{(a+b x)^2}+\frac{12 b^2 d (b c-a d)^2}{(a+b x)^3}-\frac{3 b^2 (b c-a d)^3}{(a+b x)^4}+\frac{158 b^2 d^3}{a+b x}+\frac{7 d^4 (b c-a d)}{(c+d x)^2}+\frac{98 b d^4}{c+d x}\right )}{21 (b c-a d)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(9/2)*(c + d*x)^(5/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*((-3*b^2*(b*c - a*d)^3)/(a + b*x)^4 + (12*b^2*d*(
b*c - a*d)^2)/(a + b*x)^3 - (37*b^2*d^2*(b*c - a*d))/(a + b*x)^2 + (158*b^2*d^3)
/(a + b*x) + (7*d^4*(b*c - a*d))/(c + d*x)^2 + (98*b*d^4)/(c + d*x)))/(21*(b*c -
a*d)^6)

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Maple [B]  time = 0.019, size = 356, normalized size = 1.7 \[ -{\frac{-512\,{b}^{5}{d}^{5}{x}^{5}-1792\,a{b}^{4}{d}^{5}{x}^{4}-768\,{b}^{5}c{d}^{4}{x}^{4}-2240\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}-2688\,a{b}^{4}c{d}^{4}{x}^{3}-192\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}-1120\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}-3360\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}-672\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}+32\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}-140\,{a}^{4}b{d}^{5}x-1680\,{a}^{3}{b}^{2}c{d}^{4}x-840\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x+112\,a{b}^{4}{c}^{3}{d}^{2}x-12\,{b}^{5}{c}^{4}dx+14\,{a}^{5}{d}^{5}-210\,{a}^{4}bc{d}^{4}-420\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}+140\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}-42\,a{b}^{4}{c}^{4}d+6\,{b}^{5}{c}^{5}}{21\,{d}^{6}{a}^{6}-126\,b{d}^{5}c{a}^{5}+315\,{b}^{2}{d}^{4}{c}^{2}{a}^{4}-420\,{b}^{3}{d}^{3}{c}^{3}{a}^{3}+315\,{b}^{4}{d}^{2}{c}^{4}{a}^{2}-126\,{b}^{5}d{c}^{5}a+21\,{b}^{6}{c}^{6}} \left ( bx+a \right ) ^{-{\frac{7}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(9/2)/(d*x+c)^(5/2),x)

[Out]

-2/21*(-256*b^5*d^5*x^5-896*a*b^4*d^5*x^4-384*b^5*c*d^4*x^4-1120*a^2*b^3*d^5*x^3
-1344*a*b^4*c*d^4*x^3-96*b^5*c^2*d^3*x^3-560*a^3*b^2*d^5*x^2-1680*a^2*b^3*c*d^4*
x^2-336*a*b^4*c^2*d^3*x^2+16*b^5*c^3*d^2*x^2-70*a^4*b*d^5*x-840*a^3*b^2*c*d^4*x-
420*a^2*b^3*c^2*d^3*x+56*a*b^4*c^3*d^2*x-6*b^5*c^4*d*x+7*a^5*d^5-105*a^4*b*c*d^4
-210*a^3*b^2*c^2*d^3+70*a^2*b^3*c^3*d^2-21*a*b^4*c^4*d+3*b^5*c^5)/(b*x+a)^(7/2)/
(d*x+c)^(3/2)/(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^
2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)

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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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.73524, size = 1349, normalized size = 6.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/2)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

2/21*(256*b^5*d^5*x^5 - 3*b^5*c^5 + 21*a*b^4*c^4*d - 70*a^2*b^3*c^3*d^2 + 210*a^
3*b^2*c^2*d^3 + 105*a^4*b*c*d^4 - 7*a^5*d^5 + 128*(3*b^5*c*d^4 + 7*a*b^4*d^5)*x^
4 + 32*(3*b^5*c^2*d^3 + 42*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3 - 16*(b^5*c^3*d^2 -
21*a*b^4*c^2*d^3 - 105*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)*x^2 + 2*(3*b^5*c^4*d - 2
8*a*b^4*c^3*d^2 + 210*a^2*b^3*c^2*d^3 + 420*a^3*b^2*c*d^4 + 35*a^4*b*d^5)*x)*sqr
t(b*x + a)*sqrt(d*x + c)/(a^4*b^6*c^8 - 6*a^5*b^5*c^7*d + 15*a^6*b^4*c^6*d^2 - 2
0*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4 - 6*a^9*b*c^3*d^5 + a^10*c^2*d^6 + (b^10*
c^6*d^2 - 6*a*b^9*c^5*d^3 + 15*a^2*b^8*c^4*d^4 - 20*a^3*b^7*c^3*d^5 + 15*a^4*b^6
*c^2*d^6 - 6*a^5*b^5*c*d^7 + a^6*b^4*d^8)*x^6 + 2*(b^10*c^7*d - 4*a*b^9*c^6*d^2
+ 3*a^2*b^8*c^5*d^3 + 10*a^3*b^7*c^4*d^4 - 25*a^4*b^6*c^3*d^5 + 24*a^5*b^5*c^2*d
^6 - 11*a^6*b^4*c*d^7 + 2*a^7*b^3*d^8)*x^5 + (b^10*c^8 + 2*a*b^9*c^7*d - 27*a^2*
b^8*c^6*d^2 + 64*a^3*b^7*c^5*d^3 - 55*a^4*b^6*c^4*d^4 - 6*a^5*b^5*c^3*d^5 + 43*a
^6*b^4*c^2*d^6 - 28*a^7*b^3*c*d^7 + 6*a^8*b^2*d^8)*x^4 + 4*(a*b^9*c^8 - 3*a^2*b^
8*c^7*d - 2*a^3*b^7*c^6*d^2 + 19*a^4*b^6*c^5*d^3 - 30*a^5*b^5*c^4*d^4 + 19*a^6*b
^4*c^3*d^5 - 2*a^7*b^3*c^2*d^6 - 3*a^8*b^2*c*d^7 + a^9*b*d^8)*x^3 + (6*a^2*b^8*c
^8 - 28*a^3*b^7*c^7*d + 43*a^4*b^6*c^6*d^2 - 6*a^5*b^5*c^5*d^3 - 55*a^6*b^4*c^4*
d^4 + 64*a^7*b^3*c^3*d^5 - 27*a^8*b^2*c^2*d^6 + 2*a^9*b*c*d^7 + a^10*d^8)*x^2 +
2*(2*a^3*b^7*c^8 - 11*a^4*b^6*c^7*d + 24*a^5*b^5*c^6*d^2 - 25*a^6*b^4*c^5*d^3 +
10*a^7*b^3*c^4*d^4 + 3*a^8*b^2*c^3*d^5 - 4*a^9*b*c^2*d^6 + a^10*c*d^7)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.7122, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/2)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

Done