3.16.9 \(\int \frac {1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx\) [1509]

Optimal. Leaf size=136 \[ -\frac {2}{5 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}+\frac {4 d}{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {16 d^2}{5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {32 d^3 \sqrt {a+b x}}{5 (b c-a d)^4 \sqrt {c+d x}} \]

[Out]

-2/5/(-a*d+b*c)/(b*x+a)^(5/2)/(d*x+c)^(1/2)+4/5*d/(-a*d+b*c)^2/(b*x+a)^(3/2)/(d*x+c)^(1/2)-16/5*d^2/(-a*d+b*c)
^3/(b*x+a)^(1/2)/(d*x+c)^(1/2)-32/5*d^3*(b*x+a)^(1/2)/(-a*d+b*c)^4/(d*x+c)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 d^3 \sqrt {a+b x}}{5 \sqrt {c+d x} (b c-a d)^4}-\frac {16 d^2}{5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}+\frac {4 d}{5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{5 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/2)*(c + d*x)^(3/2)),x]

[Out]

-2/(5*(b*c - a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x]) + (4*d)/(5*(b*c - a*d)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (16
*d^2)/(5*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x]) - (32*d^3*Sqrt[a + b*x])/(5*(b*c - a*d)^4*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)^{7/2} (c+d x)^{3/2}} \, dx &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}-\frac {(6 d) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{5 (b c-a d)}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}+\frac {4 d}{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {\left (8 d^2\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{5 (b c-a d)^2}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}+\frac {4 d}{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {16 d^2}{5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {\left (16 d^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{5 (b c-a d)^3}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}+\frac {4 d}{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {16 d^2}{5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {32 d^3 \sqrt {a+b x}}{5 (b c-a d)^4 \sqrt {c+d x}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 93, normalized size = 0.68 \begin {gather*} -\frac {2 (c+d x)^{5/2} \left (b^3+\frac {5 d^3 (a+b x)^3}{(c+d x)^3}+\frac {15 b d^2 (a+b x)^2}{(c+d x)^2}-\frac {5 b^2 d (a+b x)}{c+d x}\right )}{5 (b c-a d)^4 (a+b x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/2)*(c + d*x)^(3/2)),x]

[Out]

(-2*(c + d*x)^(5/2)*(b^3 + (5*d^3*(a + b*x)^3)/(c + d*x)^3 + (15*b*d^2*(a + b*x)^2)/(c + d*x)^2 - (5*b^2*d*(a
+ b*x))/(c + d*x)))/(5*(b*c - a*d)^4*(a + b*x)^(5/2))

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Maple [A]
time = 0.16, size = 145, normalized size = 1.07

method	result	size
default	\(-\frac {2}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}}-\frac {6 d \left (-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}-\frac {4 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\right )}{3 \left (-a d +b c \right )}\right )}{5 \left (-a d +b c \right )}\)	\(145\)
gosper	\(-\frac {2 \left (16 b^{3} x^{3} d^{3}+40 d^{3} a \,x^{2} b^{2}+8 b^{3} c \,d^{2} x^{2}+30 a^{2} b \,d^{3} x +20 a \,b^{2} c \,d^{2} x -2 b^{3} c^{2} d x +5 a^{3} d^{3}+15 a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{5 \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}\, \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(170\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(7/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/(-a*d+b*c)/(b*x+a)^(5/2)/(d*x+c)^(1/2)-6/5*d/(-a*d+b*c)*(-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)^(7/2)/(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. 455 vs. \(2 (112) = 224\).
time = 1.59, size = 455, normalized size = 3.35 \begin {gather*} -\frac {2 \, {\left (16 \, b^{3} d^{3} x^{3} + b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 8 \, {\left (b^{3} c d^{2} + 5 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (b^{3} c^{2} d - 10 \, a b^{2} c d^{2} - 15 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{5 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(16*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 5*a^3*d^3 + 8*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2
- 2*(b^3*c^2*d - 10*a*b^2*c*d^2 - 15*a^2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^3*b^4*c^5 - 4*a^4*b^3*c^4*d
+ 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2*d^3 + a^7*c*d^4 + (b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b
^4*c*d^4 + a^4*b^3*d^5)*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d
^4 + 3*a^5*b^2*d^5)*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c
*d^4 + a^6*b*d^5)*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d
^4 + a^7*d^5)*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 {7}{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)**(7/2)/(d*x+c)**(3/2),x)

[Out]

Integral(1/((a + b*x)**(7/2)*(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. 830 vs. \(2 (112) = 224\).
time = 2.80, size = 830, normalized size = 6.10 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b^{2} d^{3}}{{\left (b^{4} c^{4} {\left | b \right |} - 4 \, a b^{3} c^{3} d {\left | b \right |} + 6 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 4 \, a^{3} b c d^{3} {\left | b \right |} + a^{4} d^{4} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {4 \, {\left (11 \, \sqrt {b d} b^{10} c^{4} d^{2} - 44 \, \sqrt {b d} a b^{9} c^{3} d^{3} + 66 \, \sqrt {b d} a^{2} b^{8} c^{2} d^{4} - 44 \, \sqrt {b d} a^{3} b^{7} c d^{5} + 11 \, \sqrt {b d} a^{4} b^{6} d^{6} - 50 \, \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^{8} c^{3} d^{2} + 150 \, \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^{7} c^{2} d^{3} - 150 \, \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^{2} b^{6} c d^{4} + 50 \, \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^{3} b^{5} d^{5} + 80 \, \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^{6} c^{2} d^{2} - 160 \, \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} a b^{5} c d^{3} + 80 \, \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} a^{2} b^{4} d^{4} - 30 \, \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 )}^{6} b^{4} c d^{2} + 30 \, \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 )}^{6} a b^{3} d^{3} + 5 \, \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 )}^{8} b^{2} d^{2}\right )}}{5 \, {\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 )} {\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 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(b*x + a)*b^2*d^3/((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*ab
s(b) + a^4*d^4*abs(b))*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) - 4/5*(11*sqrt(b*d)*b^10*c^4*d^2 - 44*sqrt(b*d)*a*
b^9*c^3*d^3 + 66*sqrt(b*d)*a^2*b^8*c^2*d^4 - 44*sqrt(b*d)*a^3*b^7*c*d^5 + 11*sqrt(b*d)*a^4*b^6*d^6 - 50*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^8*c^3*d^2 + 150*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7*c^2*d^3 - 150*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^6*c*d^4 + 50*sqrt(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^5*d^5 + 80*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^4*b^6*c^2*d^2 - 160*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^5*c*
d^3 + 80*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^4*d^4 - 30*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^4*c*d^2 + 30*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^3*d^3 + 5*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^8*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))*(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)^5)

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Mupad [B]
time = 1.31, size = 227, normalized size = 1.67 \begin {gather*} -\frac {\sqrt {c+d\,x}\,\left (\frac {16\,d\,x^2\,\left (5\,a\,d+b\,c\right )}{5\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,a^3\,d^3+6\,a^2\,b\,c\,d^2-2\,a\,b^2\,c^2\,d+\frac {2\,b^3\,c^3}{5}}{b^2\,d\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,b\,d^2\,x^3}{5\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,x\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d-b^2\,c^2\right )}{5\,b\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a^2\,c\,\sqrt {a+b\,x}}{b^2\,d}+\frac {x^2\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {a\,x\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b^2\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(7/2)*(c + d*x)^(3/2)),x)

[Out]

-((c + d*x)^(1/2)*((16*d*x^2*(5*a*d + b*c))/(5*(a*d - b*c)^4) + (2*a^3*d^3 + (2*b^3*c^3)/5 - 2*a*b^2*c^2*d + 6
*a^2*b*c*d^2)/(b^2*d*(a*d - b*c)^4) + (32*b*d^2*x^3)/(5*(a*d - b*c)^4) + (4*x*(15*a^2*d^2 - b^2*c^2 + 10*a*b*c
*d))/(5*b*(a*d - b*c)^4)))/(x^3*(a + b*x)^(1/2) + (a^2*c*(a + b*x)^(1/2))/(b^2*d) + (x^2*(2*a*d + b*c)*(a + b*
x)^(1/2))/(b*d) + (a*x*(a*d + 2*b*c)*(a + b*x)^(1/2))/(b^2*d))

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