3.16.8 \(\int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx\) [1508]

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

[Out]

-2/3/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(1/2)+8/3*d/(-a*d+b*c)^2/(b*x+a)^(1/2)/(d*x+c)^(1/2)+16/3*d^2*(b*x+a)^(1
/2)/(-a*d+b*c)^3/(d*x+c)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 = {47, 37} \begin {gather*} \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)} \end {gather*}

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 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 47

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])

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*}

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Mathematica [A]
time = 0.10, size = 71, normalized size = 0.70 \begin {gather*} -\frac {2 (c+d x)^{3/2} \left (b^2-\frac {3 d^2 (a+b x)^2}{(c+d x)^2}-\frac {6 b d (a+b x)}{c+d x}\right )}{3 (b c-a d)^3 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [A]
time = 0.16, size = 105, normalized size = 1.04

method	result	size
gosper	\(-\frac {2 \left (8 b^{2} x^{2} d^{2}+12 a b \,d^{2} x +4 b^{2} c d x +3 a^{2} d^{2}+6 a b c d -b^{2} c^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(105\)
default	\(-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}-\frac {4 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\right )}{3 \left (-a d +b c \right )}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (83) = 166\).
time = 0.41, size = 273, normalized size = 2.70 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (83) = 166\).
time = 0.07, size = 433, normalized size = 4.29 \begin {gather*} 2 \left (\frac {2 b^{2} d^{2} \sqrt {a+b x} \sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}}{\left (2 b^{3} c^{3} \left |b\right |-6 b^{2} d a c^{2} \left |b\right |+6 b d^{2} a^{2} c \left |b\right |-2 d^{3} a^{3} \left |b\right |\right ) \left (-a b d+b^{2} c+b d \left (a+b x\right )\right )}+\frac {2 \left (-3 b^{2} d \sqrt {b d} \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{4}-12 b^{3} d^{2} \sqrt {b d} \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2} a+12 b^{4} d \sqrt {b d} \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2} c-5 b^{4} d^{3} \sqrt {b d} a^{2}+10 b^{5} d^{2} \sqrt {b d} a c-5 b^{6} d \sqrt {b d} c^{2}\right )}{\left (3 d^{2} a^{2} \left |b\right |-6 b d a c \left |b\right |+3 b^{2} c^{2} \left |b\right |\right ) \left (\left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2}+b d a-b^{2} c\right )^{3}}\right ) \end {gather*}

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]

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)

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Mupad [B]
time = 1.06, size = 141, normalized size = 1.40 \begin {gather*} -\frac {\sqrt {c+d\,x}\,\left (\frac {8\,x\,\left (3\,a\,d+b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,b\,d\,x^2}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {6\,a^2\,d^2+12\,a\,b\,c\,d-2\,b^2\,c^2}{3\,b\,d\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a\,c\,\sqrt {a+b\,x}}{b\,d}+\frac {x\,\left (a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c + d*x)^(1/2)*((8*x*(3*a*d + b*c))/(3*(a*d - b*c)^3) + (16*b*d*x^2)/(3*(a*d - b*c)^3) + (6*a^2*d^2 - 2*b^2
*c^2 + 12*a*b*c*d)/(3*b*d*(a*d - b*c)^3)))/(x^2*(a + b*x)^(1/2) + (a*c*(a + b*x)^(1/2))/(b*d) + (x*(a*d + b*c)
*(a + b*x)^(1/2))/(b*d))

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