3.15.100 \(\int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx\) [1500]

Optimal. Leaf size=136 \[ -\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 \sqrt {c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}+\frac {32 d^3 \sqrt {c+d x}}{35 (b c-a d)^4 \sqrt {a+b x}} \]

[Out]

-2/7*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(7/2)+12/35*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/(b*x+a)^(5/2)-16/35*d^2*(d*x+c)
^(1/2)/(-a*d+b*c)^3/(b*x+a)^(3/2)+32/35*d^3*(d*x+c)^(1/2)/(-a*d+b*c)^4/(b*x+a)^(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 {c+d x}}{35 \sqrt {a+b x} (b c-a d)^4}-\frac {16 d^2 \sqrt {c+d x}}{35 (a+b x)^{3/2} (b c-a d)^3}+\frac {12 d \sqrt {c+d x}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{7 (a+b x)^{7/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(9/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x])/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (12*d*Sqrt[c + d*x])/(35*(b*c - a*d)^2*(a + b*x)^(5/2)) -
(16*d^2*Sqrt[c + d*x])/(35*(b*c - a*d)^3*(a + b*x)^(3/2)) + (32*d^3*Sqrt[c + d*x])/(35*(b*c - a*d)^4*Sqrt[a +
b*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)^{9/2} \sqrt {c+d x}} \, dx &=-\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}-\frac {(6 d) \int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx}{7 (b c-a d)}\\ &=-\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}+\frac {\left (24 d^2\right ) \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx}{35 (b c-a d)^2}\\ &=-\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 \sqrt {c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}-\frac {\left (16 d^3\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{35 (b c-a d)^3}\\ &=-\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 \sqrt {c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}+\frac {32 d^3 \sqrt {c+d x}}{35 (b c-a d)^4 \sqrt {a+b x}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 93, normalized size = 0.68 \begin {gather*} -\frac {2 \sqrt {c+d x} \left (-35 d^3 (a+b x)^3+35 b d^2 (a+b x)^2 (c+d x)-21 b^2 d (a+b x) (c+d x)^2+5 b^3 (c+d x)^3\right )}{35 (b c-a d)^4 (a+b x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(9/2)*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[c + d*x]*(-35*d^3*(a + b*x)^3 + 35*b*d^2*(a + b*x)^2*(c + d*x) - 21*b^2*d*(a + b*x)*(c + d*x)^2 + 5*b
^3*(c + d*x)^3))/(35*(b*c - a*d)^4*(a + b*x)^(7/2))

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Maple [A]
time = 0.20, size = 135, normalized size = 0.99

method	result	size
default	\(-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\)	\(135\)
gosper	\(\frac {2 \sqrt {d x +c}\, \left (16 b^{3} x^{3} d^{3}+56 d^{3} a \,x^{2} b^{2}-8 b^{3} c \,d^{2} x^{2}+70 a^{2} b \,d^{3} x -28 a \,b^{2} c \,d^{2} x +6 b^{3} c^{2} d x +35 a^{3} d^{3}-35 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right )}{35 \left (b x +a \right )^{\frac {7}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(9/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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)^(9/2)/(d*x+c)^(1/2),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. 419 vs. \(2 (112) = 224\).
time = 1.56, size = 419, normalized size = 3.08 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{3} x^{3} - 5 \, b^{3} c^{3} + 21 \, a b^{2} c^{2} d - 35 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 8 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (3 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{35 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/35*(16*b^3*d^3*x^3 - 5*b^3*c^3 + 21*a*b^2*c^2*d - 35*a^2*b*c*d^2 + 35*a^3*d^3 - 8*(b^3*c*d^2 - 7*a*b^2*d^3)*
x^2 + 2*(3*b^3*c^2*d - 14*a*b^2*c*d^2 + 35*a^2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b^4*c^4 - 4*a^5*b^3*
c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5
*c*d^3 + a^4*b^4*d^4)*x^4 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4
)*x^3 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^3*b^5
*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*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 {9}{2}} \sqrt {c + d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(9/2)/(d*x+c)**(1/2),x)

[Out]

Integral(1/((a + b*x)**(9/2)*sqrt(c + d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (112) = 224\).
time = 1.22, size = 386, normalized size = 2.84 \begin {gather*} \frac {64 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 7 \, {\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^{2} + 14 \, {\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} c d - 7 \, {\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^{2} d^{2} + 21 \, {\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} c - 21 \, {\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 d - 35 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6}\right )} \sqrt {b d} b^{4} d^{3}}{35 \, {\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 )}^{7} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

64/35*(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3 - 7*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^2*b^4*c^2 + 14*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^3*c*
d - 7*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^2*d^2 + 21*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c - 21*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4*a*b*d - 35*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6)*sqrt(b*d)*b^4*d^3/((
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)^7*abs(b))

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Mupad [B]
time = 1.19, size = 209, normalized size = 1.54 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {32\,d^3\,x^3}{35\,{\left (a\,d-b\,c\right )}^4}+\frac {70\,a^3\,d^3-70\,a^2\,b\,c\,d^2+42\,a\,b^2\,c^2\,d-10\,b^3\,c^3}{35\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d\,x\,\left (35\,a^2\,d^2-14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{35\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^2\,x^2\,\left (7\,a\,d-b\,c\right )}{35\,b\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a^3\,\sqrt {a+b\,x}}{b^3}+\frac {3\,a\,x^2\,\sqrt {a+b\,x}}{b}+\frac {3\,a^2\,x\,\sqrt {a+b\,x}}{b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(9/2)*(c + d*x)^(1/2)),x)

[Out]

((c + d*x)^(1/2)*((32*d^3*x^3)/(35*(a*d - b*c)^4) + (70*a^3*d^3 - 10*b^3*c^3 + 42*a*b^2*c^2*d - 70*a^2*b*c*d^2
)/(35*b^3*(a*d - b*c)^4) + (4*d*x*(35*a^2*d^2 + 3*b^2*c^2 - 14*a*b*c*d))/(35*b^2*(a*d - b*c)^4) + (16*d^2*x^2*
(7*a*d - b*c))/(35*b*(a*d - b*c)^4)))/(x^3*(a + b*x)^(1/2) + (a^3*(a + b*x)^(1/2))/b^3 + (3*a*x^2*(a + b*x)^(1
/2))/b + (3*a^2*x*(a + b*x)^(1/2))/b^2)

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