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])

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Rubi [A]  time = 0.0441092, 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, Rules used = {45, 37} \[ -\frac{256 b d^3 \sqrt{a+b x}}{15 \sqrt{c+d x} (b c-a d)^5}-\frac{128 d^3 \sqrt{a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac{16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]

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])

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)^{7/2} (c+d x)^{5/2}} \, dx &=-\frac{2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}-\frac{(8 d) \int \frac{1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{5 (b c-a d)}\\ &=-\frac{2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac{16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac{\left (16 d^2\right ) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{5 (b c-a d)^2}\\ &=-\frac{2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac{16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{32 d^2}{5 (b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}-\frac{\left (64 d^3\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx}{5 (b c-a d)^3}\\ &=-\frac{2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac{16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{32 d^2}{5 (b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}-\frac{128 d^3 \sqrt{a+b x}}{15 (b c-a d)^4 (c+d x)^{3/2}}-\frac{\left (128 b d^3\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{15 (b c-a d)^4}\\ &=-\frac{2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac{16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac{32 d^2}{5 (b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}-\frac{128 d^3 \sqrt{a+b x}}{15 (b c-a d)^4 (c+d x)^{3/2}}-\frac{256 b d^3 \sqrt{a+b x}}{15 (b c-a d)^5 \sqrt{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0622627, size = 170, normalized size = 0.99 \[ -\frac{2 \left (30 a^2 b^2 d^2 \left (3 c^2+12 c d x+8 d^2 x^2\right )+20 a^3 b d^3 (3 c+2 d x)-5 a^4 d^4+20 a b^3 d \left (6 c^2 d x-c^3+24 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (48 c^2 d^2 x^2-8 c^3 d x+3 c^4+192 c d^3 x^3+128 d^4 x^4\right )\right )}{15 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-5*a^4*d^4 + 20*a^3*b*d^3*(3*c + 2*d*x) + 30*a^2*b^2*d^2*(3*c^2 + 12*c*d*x + 8*d^2*x^2) + 20*a*b^3*d*(-c^
3 + 6*c^2*d*x + 24*c*d^2*x^2 + 16*d^3*x^3) + b^4*(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)))/(15*(b*c - a*d)^5*(a + b*x)^(5/2)*(c + d*x)^(3/2))

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Maple [A]  time = 0.01, size = 256, normalized size = 1.5 \begin{align*} -{\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}}}} \end{align*}

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)

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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)^(7/2)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 43.6118, size = 1440, normalized size = 8.37 \begin{align*} -\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 )}} \end{align*}

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 + 10*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)

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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)**(7/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 3.22649, size = 1362, normalized size = 7.92 \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)^(7/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

1/24*sqrt(b*x + a)*(11*(b^8*c^4*d^5*abs(b) - 4*a*b^7*c^3*d^6*abs(b) + 6*a^2*b^6*c^2*d^7*abs(b) - 4*a^3*b^5*c*d
^8*abs(b) + a^4*b^4*d^9*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) + 12*(b^9*c^5*d^4*abs(b)
- 5*a*b^8*c^4*d^5*abs(b) + 10*a^2*b^7*c^3*d^6*abs(b) - 10*a^3*b^6*c^2*d^7*abs(b) + 5*a^4*b^5*c*d^8*abs(b) - a
^5*b^4*d^9*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) - 4/15*(
73*sqrt(b*d)*b^11*c^4*d^2 - 292*sqrt(b*d)*a*b^10*c^3*d^3 + 438*sqrt(b*d)*a^2*b^9*c^2*d^4 - 292*sqrt(b*d)*a^3*b
^8*c*d^5 + 73*sqrt(b*d)*a^4*b^7*d^6 - 320*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^2*b^9*c^3*d^2 + 960*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^8*c^
2*d^3 - 960*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^7*c*d^4 + 320*sq
rt(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^6*d^5 + 490*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^7*c^2*d^2 - 980*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^6*c*d^3 + 490*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^5*d^4 - 240*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^6*b^5*c*d^2 + 240*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^4*d^
3 + 45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^3*d^2)/((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 - (sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^5)