3.789 \(\int \frac{1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{16 b d \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}-\frac{8 d \sqrt{a+b x}}{3 (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]

[Out]

-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (8*d*Sqrt[a + b*x])/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) - (16*b
*d*Sqrt[a + b*x])/(3*(b*c - a*d)^3*Sqrt[c + d*x])

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Rubi [A]  time = 0.0197125, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{16 b d \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}-\frac{8 d \sqrt{a+b x}}{3 (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (8*d*Sqrt[a + b*x])/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) - (16*b
*d*Sqrt[a + b*x])/(3*(b*c - a*d)^3*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)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{(4 d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx}{b c-a d}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{8 d \sqrt{a+b x}}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac{(8 b d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}-\frac{8 d \sqrt{a+b x}}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac{16 b d \sqrt{a+b x}}{3 (b c-a d)^3 \sqrt{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0232334, size = 78, normalized size = 0.8 \[ \frac{2 a^2 d^2-4 a b d (3 c+2 d x)-2 b^2 \left (3 c^2+12 c d x+8 d^2 x^2\right )}{3 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(2*a^2*d^2 - 4*a*b*d*(3*c + 2*d*x) - 2*b^2*(3*c^2 + 12*c*d*x + 8*d^2*x^2))/(3*(b*c - a*d)^3*Sqrt[a + b*x]*(c +
d*x)^(3/2))

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Maple [A]  time = 0.005, size = 104, normalized size = 1.1 \begin{align*} -{\frac{-16\,{b}^{2}{d}^{2}{x}^{2}-8\,ab{d}^{2}x-24\,{b}^{2}cdx+2\,{a}^{2}{d}^{2}-12\,abcd-6\,{b}^{2}{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}cb{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}}{\frac{1}{\sqrt{bx+a}}} \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)^(3/2)/(d*x+c)^(5/2),x)

[Out]

-2/3*(-8*b^2*d^2*x^2-4*a*b*d^2*x-12*b^2*c*d*x+a^2*d^2-6*a*b*c*d-3*b^2*c^2)/(b*x+a)^(1/2)/(d*x+c)^(3/2)/(a^3*d^
3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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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)^(3/2)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.20198, size = 545, normalized size = 5.56 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2} + 4 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(8*b^2*d^2*x^2 + 3*b^2*c^2 + 6*a*b*c*d - a^2*d^2 + 4*(3*b^2*c*d + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)
/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d
^4 - a^3*b*d^5)*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^2 + (b^4*c
^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(5/2)), x)

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Giac [B]  time = 1.41713, size = 393, normalized size = 4.01 \begin{align*} -\frac{4 \, \sqrt{b d} b^{3}}{{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac{\sqrt{b x + a}{\left (\frac{5 \,{\left (b^{6} c^{2} d^{3}{\left | b \right |} - 2 \, a b^{5} c d^{4}{\left | b \right |} + a^{2} b^{4} d^{5}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{6 \,{\left (b^{7} c^{3} d^{2}{\left | b \right |} - 3 \, a b^{6} c^{2} d^{3}{\left | b \right |} + 3 \, a^{2} b^{5} c d^{4}{\left | b \right |} - a^{3} b^{4} d^{5}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

-4*sqrt(b*d)*b^3/((b^2*c^2*abs(b) - 2*a*b*c*d*abs(b) + a^2*d^2*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)) + 1/24*sqrt(b*x + a)*(5*(b^6*c^2*d^3*abs(b) - 2*a*b^5*c*d^4*abs(
b) + a^2*b^4*d^5*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) + 6*(b^7*c^3*d^2*abs(b) - 3*a*b
^6*c^2*d^3*abs(b) + 3*a^2*b^5*c*d^4*abs(b) - a^3*b^4*d^5*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)