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

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

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Rubi [A]  time = 0.03, 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 = {45, 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 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])

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 = 0.05, size = 116, normalized size = 0.85 \begin {gather*} \frac {2 \sqrt {c+d x} \left (35 a^3 d^3-35 a^2 b d^2 (c-2 d x)+7 a b^2 d \left (3 c^2-4 c d x+8 d^2 x^2\right )+b^3 \left (-5 c^3+6 c^2 d x-8 c d^2 x^2+16 d^3 x^3\right )\right )}{35 (a+b x)^{7/2} (b c-a d)^4} \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*a^3*d^3 - 35*a^2*b*d^2*(c - 2*d*x) + 7*a*b^2*d*(3*c^2 - 4*c*d*x + 8*d^2*x^2) + b^3*(-5*c^
3 + 6*c^2*d*x - 8*c*d^2*x^2 + 16*d^3*x^3)))/(35*(b*c - a*d)^4*(a + b*x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.11, size = 109, normalized size = 0.80 \begin {gather*} -\frac {2 \left (\frac {5 b^3 (c+d x)^{7/2}}{(a+b x)^{7/2}}-\frac {21 b^2 d (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {35 d^3 \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {35 b d^2 (c+d x)^{3/2}}{(a+b x)^{3/2}}\right )}{35 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.89, 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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giac [B]  time = 1.47, 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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maple [A]  time = 0.01, size = 171, normalized size = 1.26 \begin {gather*} \frac {2 \sqrt {d x +c}\, \left (16 b^{3} x^{3} d^{3}+56 a \,b^{2} d^{3} x^{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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(d*x+c)^(1/2)*(16*b^3*d^3*x^3+56*a*b^2*d^3*x^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)/(b*x+a)^(7/2)/(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)

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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 details)Is a*d-b*c zero or nonzero?

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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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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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