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 = 0.0610089, 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, Rules used = {45, 37} \[ \frac{512 b d^4 \sqrt{a+b x}}{21 \sqrt{c+d x} (b c-a d)^6}+\frac{256 d^4 \sqrt{a+b x}}{21 (c+d x)^{3/2} (b c-a d)^5}+\frac{64 d^3}{7 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{21 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^3}+\frac{4 d}{7 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)} \]

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])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}-\frac{(10 d) \int \frac{1}{(a+b x)^{7/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac{4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}+\frac{\left (16 d^2\right ) \int \frac{1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^2}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac{4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac{32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{\left (32 d^3\right ) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^3}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac{4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac{32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{64 d^3}{7 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}+\frac{\left (128 d^4\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^4}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac{4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac{32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{64 d^3}{7 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}+\frac{256 d^4 \sqrt{a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac{\left (256 b d^4\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{21 (b c-a d)^5}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac{4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac{32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{64 d^3}{7 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}+\frac{256 d^4 \sqrt{a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac{512 b d^4 \sqrt{a+b x}}{21 (b c-a d)^6 \sqrt{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0785279, size = 233, normalized size = 1.13 \[ \frac{2 \left (70 a^2 b^3 d^2 \left (6 c^2 d x-c^3+24 c d^2 x^2+16 d^3 x^3\right )+70 a^3 b^2 d^3 \left (3 c^2+12 c d x+8 d^2 x^2\right )+35 a^4 b d^4 (3 c+2 d x)-7 a^5 d^5+7 a b^4 d \left (48 c^2 d^2 x^2-8 c^3 d x+3 c^4+192 c d^3 x^3+128 d^4 x^4\right )+b^5 \left (-16 c^3 d^2 x^2+96 c^2 d^3 x^3+6 c^4 d x-3 c^5+384 c d^4 x^4+256 d^5 x^5\right )\right )}{21 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-7*a^5*d^5 + 35*a^4*b*d^4*(3*c + 2*d*x) + 70*a^3*b^2*d^3*(3*c^2 + 12*c*d*x + 8*d^2*x^2) + 70*a^2*b^3*d^2*(
-c^3 + 6*c^2*d*x + 24*c*d^2*x^2 + 16*d^3*x^3) + 7*a*b^4*d*(3*c^4 - 8*c^3*d*x + 48*c^2*d^2*x^2 + 192*c*d^3*x^3
+ 128*d^4*x^4) + b^5*(-3*c^5 + 6*c^4*d*x - 16*c^3*d^2*x^2 + 96*c^2*d^3*x^3 + 384*c*d^4*x^4 + 256*d^5*x^5)))/(2
1*(b*c - a*d)^6*(a + b*x)^(7/2)*(c + d*x)^(3/2))

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Maple [B]  time = 0.011, size = 356, normalized size = 1.7 \begin{align*} -{\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}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 [B]  time = 97.4435, size = 2045, normalized size = 9.88 \begin{align*} \text{result too large to display} \end{align*}

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 - 28*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)*sqrt(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 - 20*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. \begin{align*} \text{Timed out} \end{align*}

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 [B]  time = 6.44747, size = 2314, normalized size = 11.18 \begin{align*} \text{result too large to display} \end{align*}

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]

-1/24*sqrt(b*x + a)*(14*(b^9*c^5*d^6*abs(b) - 5*a*b^8*c^4*d^7*abs(b) + 10*a^2*b^7*c^3*d^8*abs(b) - 10*a^3*b^6*
c^2*d^9*abs(b) + 5*a^4*b^5*c*d^10*abs(b) - a^5*b^4*d^11*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b
^6*d^6) + 15*(b^10*c^6*d^5*abs(b) - 6*a*b^9*c^5*d^6*abs(b) + 15*a^2*b^8*c^4*d^7*abs(b) - 20*a^3*b^7*c^3*d^8*ab
s(b) + 15*a^4*b^6*c^2*d^9*abs(b) - 6*a^5*b^5*c*d^10*abs(b) + a^6*b^4*d^11*abs(b))/(b^8*c^2*d^4 - 2*a*b^7*c*d^5
+ a^2*b^6*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 8/21*(79*sqrt(b*d)*b^15*c^6*d^3 - 474*sqrt(b*d)*a*b^1
4*c^5*d^4 + 1185*sqrt(b*d)*a^2*b^13*c^4*d^5 - 1580*sqrt(b*d)*a^3*b^12*c^3*d^6 + 1185*sqrt(b*d)*a^4*b^11*c^2*d^
7 - 474*sqrt(b*d)*a^5*b^10*c*d^8 + 79*sqrt(b*d)*a^6*b^9*d^9 - 511*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^2*b^13*c^5*d^3 + 2555*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^12*c^4*d^4 - 5110*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^11*c^3*d^5 + 5110*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^10*c^2*d^6 - 2555*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^9*c*d^7
+ 511*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^8*d^8 + 1344*sqrt(b*d)
*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^11*c^4*d^3 - 5376*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^10*c^3*d^4 + 8064*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^9*c^2*d^5 - 5376*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^4*a^3*b^8*c*d^6 + 1344*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^4*a^4*b^7*d^7 - 1750*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
6*b^9*c^3*d^3 + 5250*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^8*c^2*d^4
- 5250*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^7*c*d^5 + 1750*sqrt(
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^6*d^6 + 1015*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^7*c^2*d^3 - 2030*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^6*c*d^4 + 1015*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^8*a^2*b^5*d^5 - 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^10*b^5*c*d^3 + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^4*
d^4 + 42*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^3*d^3)/((b^5*c^5*abs(b
) - 5*a*b^4*c^4*d*abs(b) + 10*a^2*b^3*c^3*d^2*abs(b) - 10*a^3*b^2*c^2*d^3*abs(b) + 5*a^4*b*c*d^4*abs(b) - a^5*
d^5*abs(b))*(b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^7)