3.1508 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{16 d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{8 d}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0178873, antiderivative size = 101, 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} \[ \frac{16 d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{8 d}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.029055, size = 75, normalized size = 0.74 \[ \frac{2 \left (3 a^2 d^2+6 a b d (c+2 d x)+b^2 \left (-c^2+4 c d x+8 d^2 x^2\right )\right )}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*a^2*d^2 + 6*a*b*d*(c + 2*d*x) + b^2*(-c^2 + 4*c*d*x + 8*d^2*x^2)))/(3*(b*c - a*d)^3*(a + b*x)^(3/2)*Sqrt
[c + d*x])

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Maple [A]  time = 0.006, size = 105, normalized size = 1. \begin{align*} -{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+24\,ab{d}^{2}x+8\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}+12\,abcd-2\,{b}^{2}{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}bc{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(8*b^2*d^2*x^2+12*a*b*d^2*x+4*b^2*c*d*x+3*a^2*d^2+6*a*b*c*d-b^2*c^2)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(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)^(5/2)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.70748, size = 544, normalized size = 5.39 \begin{align*} \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} +{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} +{\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} +{\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(8*b^2*d^2*x^2 - b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2 + 4*(b^2*c*d + 3*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)/
(a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3 + (b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3
- a^3*b^2*d^4)*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*x^2 + (2*a*b^
4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*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{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.32288, size = 497, normalized size = 4.92 \begin{align*} \frac{2 \, \sqrt{b x + a} b^{2} d^{2}}{{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{4 \,{\left (5 \, \sqrt{b d} b^{6} c^{2} d - 10 \, \sqrt{b d} a b^{5} c d^{2} + 5 \, \sqrt{b d} a^{2} b^{4} d^{3} - 12 \, \sqrt{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} b^{4} c d + 12 \, \sqrt{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} a b^{3} d^{2} + 3 \, \sqrt{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 )}^{4} b^{2} d\right )}}{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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)*b^2*d^2/((b^3*c^3*abs(b) - 3*a*b^2*c^2*d*abs(b) + 3*a^2*b*c*d^2*abs(b) - a^3*d^3*abs(b))*sqrt(
b^2*c + (b*x + a)*b*d - a*b*d)) + 4/3*(5*sqrt(b*d)*b^6*c^2*d - 10*sqrt(b*d)*a*b^5*c*d^2 + 5*sqrt(b*d)*a^2*b^4*
d^3 - 12*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^4*c*d + 12*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^3*d^2 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*d)/((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)^3)