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*Sqrt[a + b*x])/(35*(b*c - a*d)^5*Sqrt[c + d*x])

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Rubi [A]  time = 0.0432695, 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, Rules used = {45, 37} \[ \frac{256 d^4 \sqrt{a+b x}}{35 \sqrt{c+d x} (b c-a d)^5}+\frac{128 d^3}{35 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}-\frac{32 d^2}{35 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3}+\frac{16 d}{35 (a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{7 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)} \]

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*Sqrt[a + b*x])/(35*(b*c - a*d)^5*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)^{3/2}} \, dx &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}-\frac{(8 d) \int \frac{1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx}{7 (b c-a d)}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}+\frac{16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}+\frac{\left (48 d^2\right ) \int \frac{1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{35 (b c-a d)^2}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}+\frac{16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}-\frac{32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}-\frac{\left (64 d^3\right ) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{35 (b c-a d)^3}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}+\frac{16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}-\frac{32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}+\frac{128 d^3}{35 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}+\frac{\left (128 d^4\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{35 (b c-a d)^4}\\ &=-\frac{2}{7 (b c-a d) (a+b x)^{7/2} \sqrt{c+d x}}+\frac{16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt{c+d x}}-\frac{32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt{c+d x}}+\frac{128 d^3}{35 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}+\frac{256 d^4 \sqrt{a+b x}}{35 (b c-a d)^5 \sqrt{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0578715, size = 166, normalized size = 0.97 \[ \frac{2 \left (70 a^2 b^2 d^2 \left (-c^2+4 c d x+8 d^2 x^2\right )+140 a^3 b d^3 (c+2 d x)+35 a^4 d^4+28 a b^3 d \left (-2 c^2 d x+c^3+8 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (-16 c^2 d^2 x^2+8 c^3 d x-5 c^4+64 c d^3 x^3+128 d^4 x^4\right )\right )}{35 (a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(35*a^4*d^4 + 140*a^3*b*d^3*(c + 2*d*x) + 70*a^2*b^2*d^2*(-c^2 + 4*c*d*x + 8*d^2*x^2) + 28*a*b^3*d*(c^3 - 2
*c^2*d*x + 8*c*d^2*x^2 + 16*d^3*x^3) + b^4*(-5*c^4 + 8*c^3*d*x - 16*c^2*d^2*x^2 + 64*c*d^3*x^3 + 128*d^4*x^4))
)/(35*(b*c - a*d)^5*(a + b*x)^(7/2)*Sqrt[c + d*x])

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Maple [A]  time = 0.009, size = 256, normalized size = 1.5 \begin{align*} -{\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}}}} \end{align*}

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+224*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(-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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 28.4017, size = 1400, normalized size = 8.19 \begin{align*} \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 )}} \end{align*}

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 + 1
0*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. \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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 2.83831, size = 2049, normalized size = 11.98 \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)^(3/2),x, algorithm="giac")

[Out]

2*sqrt(b*x + a)*b^2*d^4/((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))*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) + 4/35*(93*sqrt(b*d)*b
^14*c^6*d^3 - 558*sqrt(b*d)*a*b^13*c^5*d^4 + 1395*sqrt(b*d)*a^2*b^12*c^4*d^5 - 1860*sqrt(b*d)*a^3*b^11*c^3*d^6
+ 1395*sqrt(b*d)*a^4*b^10*c^2*d^7 - 558*sqrt(b*d)*a^5*b^9*c*d^8 + 93*sqrt(b*d)*a^6*b^8*d^9 - 616*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^12*c^5*d^3 + 3080*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^11*c^4*d^4 - 6160*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^10*c^3*d^5 + 6160*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^9*c^2*d^6 - 3080*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^8*c*d^7 + 616*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^7*d^8 + 1673*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^10*c^4*d^3
- 6692*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^9*c^3*d^4 + 10038*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^8*c^2*d^5 - 6692*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^7*c*d^6 + 1673*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^6*d^7 - 2240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^6*b^8*c^3*d^3 + 6720*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^7*c^2*d^4 - 6720*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^6*c*d^5 + 2240*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^5*
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^6*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^5*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^4*d^5 - 280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^4*c*d^3 + 280*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^3*d^4 + 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^12*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))*(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)