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

Optimal. Leaf size=118 \[ \frac {2 a}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {4 \sqrt {a+b x} (3 a d+b c)}{3 \sqrt {c+d x} (b c-a d)^3}+\frac {2 \sqrt {a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \frac {2 a}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {4 \sqrt {a+b x} (3 a d+b c)}{3 \sqrt {c+d x} (b c-a d)^3}+\frac {2 \sqrt {a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 83, normalized size = 0.70 \begin {gather*} \frac {2 \left (a^2 d (2 c+3 d x)+2 a b \left (3 c^2+5 c d x+3 d^2 x^2\right )+b^2 c x (3 c+2 d x)\right )}{3 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2*c*x*(3*c + 2*d*x) + a^2*d*(2*c + 3*d*x) + 2*a*b*(3*c^2 + 5*c*d*x + 3*d^2*x^2)))/(3*(b*c - a*d)^3*Sqrt[
a + b*x]*(c + d*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.13, size = 87, normalized size = 0.74 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (\frac {3 a b (c+d x)^2}{(a+b x)^2}+\frac {3 b c (c+d x)}{a+b x}+\frac {3 a d (c+d x)}{a+b x}-c d\right )}{3 (c+d x)^{3/2} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(3/2)*(-(c*d) + (3*b*c*(c + d*x))/(a + b*x) + (3*a*d*(c + d*x))/(a + b*x) + (3*a*b*(c + d*x)^2)/(
a + b*x)^2))/(3*(b*c - a*d)^3*(c + d*x)^(3/2))

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fricas [B]  time = 2.50, size = 281, normalized size = 2.38 \begin {gather*} \frac {2 \, {\left (6 \, a b c^{2} + 2 \, a^{2} c d + 2 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x^{2} + {\left (3 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(6*a*b*c^2 + 2*a^2*c*d + 2*(b^2*c*d + 3*a*b*d^2)*x^2 + (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*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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giac [B]  time = 2.28, size = 393, normalized size = 3.33 \begin {gather*} \frac {2 \, {\left (\frac {6 \, \sqrt {b d} a 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 {{\left (2 \, b^{7} c^{3} d^{2} {\left | b \right |} - a b^{6} c^{2} d^{3} {\left | b \right |} - 4 \, a^{2} b^{5} c d^{4} {\left | b \right |} + 3 \, a^{3} b^{4} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{2} b^{5} c^{3} d^{3} - 10 \, a^{3} b^{4} c^{2} d^{4} + 5 \, a^{4} b^{3} c d^{5} - a^{5} b^{2} d^{6}} + \frac {3 \, {\left (b^{8} c^{4} d {\left | b \right |} - 2 \, a b^{7} c^{3} d^{2} {\left | b \right |} + 2 \, a^{3} b^{5} c d^{4} {\left | b \right |} - a^{4} b^{4} d^{5} {\left | b \right |}\right )}}{b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{2} b^{5} c^{3} d^{3} - 10 \, a^{3} b^{4} c^{2} d^{4} + 5 \, a^{4} b^{3} c d^{5} - a^{5} b^{2} d^{6}}\right )}}{{\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}}\right )}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(6*sqrt(b*d)*a*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)) + sqrt(b*x + a)*((2*b^7*c^3*d^2*abs(b) - a*b^6*c^2*d^3*abs
(b) - 4*a^2*b^5*c*d^4*abs(b) + 3*a^3*b^4*d^5*abs(b))*(b*x + a)/(b^7*c^5*d - 5*a*b^6*c^4*d^2 + 10*a^2*b^5*c^3*d
^3 - 10*a^3*b^4*c^2*d^4 + 5*a^4*b^3*c*d^5 - a^5*b^2*d^6) + 3*(b^8*c^4*d*abs(b) - 2*a*b^7*c^3*d^2*abs(b) + 2*a^
3*b^5*c*d^4*abs(b) - a^4*b^4*d^5*abs(b))/(b^7*c^5*d - 5*a*b^6*c^4*d^2 + 10*a^2*b^5*c^3*d^3 - 10*a^3*b^4*c^2*d^
4 + 5*a^4*b^3*c*d^5 - a^5*b^2*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2))/b

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maple [A]  time = 0.01, size = 115, normalized size = 0.97 \begin {gather*} -\frac {2 \left (6 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +10 a b c d x +3 b^{2} c^{2} x +2 a^{2} c d +6 a b \,c^{2}\right )}{3 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(6*a*b*d^2*x^2+2*b^2*c*d*x^2+3*a^2*d^2*x+10*a*b*c*d*x+3*b^2*c^2*x+2*a^2*c*d+6*a*b*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 1.67, size = 143, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,a^2\,d^2+20\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,b\,x^2\,\left (3\,a\,d+b\,c\right )}{3\,d\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,a\,c\,\left (a\,d+3\,b\,c\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c + d*x)^(1/2)*((x*(6*a^2*d^2 + 6*b^2*c^2 + 20*a*b*c*d))/(3*d^2*(a*d - b*c)^3) + (4*b*x^2*(3*a*d + b*c))/(3
*d*(a*d - b*c)^3) + (4*a*c*(a*d + 3*b*c))/(3*d^2*(a*d - b*c)^3)))/(x^2*(a + b*x)^(1/2) + (c^2*(a + b*x)^(1/2))
/d^2 + (2*c*x*(a + b*x)^(1/2))/d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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