3.1478 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(5/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35*(b*c - a*d)^2*(a + b*x)^(5/2)
)

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Rubi [A]  time = 0.0082446, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35*(b*c - a*d)^2*(a + b*x)^(5/2)
)

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 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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx &=-\frac{2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}-\frac{(2 d) \int \frac{(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{7 (b c-a d)}\\ &=-\frac{2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac{4 d (c+d x)^{5/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0250242, size = 46, normalized size = 0.7 \[ \frac{2 (c+d x)^{5/2} (7 a d-5 b c+2 b d x)}{35 (a+b x)^{7/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)/(a + b*x)^(9/2),x]

[Out]

(2*(c + d*x)^(5/2)*(-5*b*c + 7*a*d + 2*b*d*x))/(35*(b*c - a*d)^2*(a + b*x)^(7/2))

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Maple [A]  time = 0.005, size = 54, normalized size = 0.8 \begin{align*}{\frac{4\,bdx+14\,ad-10\,bc}{35\,{a}^{2}{d}^{2}-70\,abcd+35\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(3/2)/(b*x+a)^(9/2),x)

[Out]

2/35*(d*x+c)^(5/2)*(2*b*d*x+7*a*d-5*b*c)/(b*x+a)^(7/2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 11.8059, size = 483, normalized size = 7.32 \begin{align*} \frac{2 \,{\left (2 \, b d^{3} x^{3} - 5 \, b c^{3} + 7 \, a c^{2} d -{\left (b c d^{2} - 7 \, a d^{3}\right )} x^{2} - 2 \,{\left (4 \, b c^{2} d - 7 \, a c d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{35 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(3/2)/(b*x+a)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.53585, size = 1382, normalized size = 20.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^(9/2),x, algorithm="giac")

[Out]

8/35*(sqrt(b*d)*b^10*c^5*d^3*abs(b) - 5*sqrt(b*d)*a*b^9*c^4*d^4*abs(b) + 10*sqrt(b*d)*a^2*b^8*c^3*d^5*abs(b) -
10*sqrt(b*d)*a^3*b^7*c^2*d^6*abs(b) + 5*sqrt(b*d)*a^4*b^6*c*d^7*abs(b) - sqrt(b*d)*a^5*b^5*d^8*abs(b) - 7*sqr
t(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^4*d^3*abs(b) + 28*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7*c^3*d^4*abs(b) - 42*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^2*d^5*abs(b) + 28*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*c*d^6*abs(b) - 7*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^4*d^7*abs(b) - 14*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^3*d^3*abs(b) + 42*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^2*d^4*abs(b) - 42*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*c*d^5*abs(b) + 14*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*a^3*b^3*d^6*abs(b) - 70*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^
2*d^3*abs(b) + 140*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*c*d^4*abs
(b) - 70*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*b^2*d^5*abs(b) - 35*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^2*c*d^3*abs(b) + 35*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b*d^4*abs(b) - 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*d^3*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^2)^7*b^2)