3.11.24 \(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=39 \[ \frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {682} \begin {gather*} \frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]

[Out]

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

Rule 682

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx &=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.97 \begin {gather*} \frac {2 (a+x (b+c x))^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.76, size = 76, normalized size = 1.95 \begin {gather*} \frac {2 \sqrt {a+b x+c x^2} \left (a^2+2 a b x+2 a c x^2+b^2 x^2+2 b c x^3+c^2 x^4\right )}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]

[Out]

(2*Sqrt[a + b*x + c*x^2]*(a^2 + 2*a*b*x + b^2*x^2 + 2*a*c*x^2 + 2*b*c*x^3 + c^2*x^4))/(5*(b^2 - 4*a*c)*d^6*(b
+ 2*c*x)^5)

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fricas [B]  time = 2.09, size = 183, normalized size = 4.69 \begin {gather*} \frac {2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a}}{5 \, {\left (32 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \, {\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \, {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \, {\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x + {\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")

[Out]

2/5*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)/(32*(b^2*c^5 - 4*a*c^6)*d^
6*x^5 + 80*(b^3*c^4 - 4*a*b*c^5)*d^6*x^4 + 80*(b^4*c^3 - 4*a*b^2*c^4)*d^6*x^3 + 40*(b^5*c^2 - 4*a*b^3*c^3)*d^6
*x^2 + 10*(b^6*c - 4*a*b^4*c^2)*d^6*x + (b^7 - 4*a*b^5*c)*d^6)

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giac [B]  time = 0.61, size = 594, normalized size = 15.23 \begin {gather*} \frac {80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} c^{\frac {9}{2}} + 320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} b c^{4} + 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} b^{2} c^{\frac {7}{2}} + 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{3} c^{3} + 360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{4} c^{\frac {5}{2}} - 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{2} c^{\frac {7}{2}} + 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} c^{\frac {9}{2}} + 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{5} c^{2} - 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{3} c^{3} + 320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b c^{4} + 50 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{6} c^{\frac {3}{2}} - 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{4} c^{\frac {5}{2}} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{\frac {7}{2}} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{7} c - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{5} c^{2} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{3} c^{3} + b^{8} \sqrt {c} - 6 \, a b^{6} c^{\frac {3}{2}} + 16 \, a^{2} b^{4} c^{\frac {5}{2}} - 16 \, a^{3} b^{2} c^{\frac {7}{2}} + 16 \, a^{4} c^{\frac {9}{2}}}{80 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{5} c^{3} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")

[Out]

1/80*(80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(9/2) + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b*c^4 + 560
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(7/2) + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c^3 + 360*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(5/2) - 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(7/2) + 16
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(9/2) + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^2 - 160*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^3 + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^4 + 50*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c^(3/2) - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(5/2) + 240
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(7/2) + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^7*c - 40*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^2 + 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c^3 + b^8*sqrt(c)
- 6*a*b^6*c^(3/2) + 16*a^2*b^4*c^(5/2) - 16*a^3*b^2*c^(7/2) + 16*a^4*c^(9/2))/((2*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^5*c^3*d^6)

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maple [A]  time = 0.04, size = 38, normalized size = 0.97 \begin {gather*} -\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 \left (2 c x +b \right )^{5} \left (4 a c -b^{2}\right ) d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^5/d^6/(4*a*c-b^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((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,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(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 1.53, size = 1156, normalized size = 29.64 \begin {gather*} \frac {\left (\frac {b\,\left (\frac {b\,\left (\frac {b\,\left (\frac {32\,a\,c^4}{3\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}-\frac {16\,b^2\,c^3}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )}{2\,c}-\frac {8\,b\,c^2\,\left (6\,a\,c-b^2\right )}{3\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )}{2\,c}+\frac {4\,c\,\left (36\,a^2\,c^2+12\,a\,b^2\,c-4\,b^4\right )}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )}{2\,c}-\frac {8\,a\,b\,c\,\left (9\,a\,c-2\,b^2\right )}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^2\,\left (32\,a\,c^3-8\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^2}-\frac {\left (\frac {b\,\left (\frac {b\,\left (\frac {2\,c^2\,\left (28\,a\,c-b^2\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}-\frac {6\,b^2\,c^2}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}\right )}{2\,c}+\frac {2\,c\,\left (5\,b^3-28\,a\,b\,c\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}\right )}{2\,c}-\frac {2\,c\,\left (5\,a\,b^2-24\,a^2\,c\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (48\,a\,c^3-12\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^3}+\frac {\left (\frac {8\,a\,c-b^2}{40\,c^2\,d^6\,{\left (4\,a\,c-b^2\right )}^2}-\frac {b^2}{40\,c^2\,d^6\,{\left (4\,a\,c-b^2\right )}^2}\right )\,\sqrt {c\,x^2+b\,x+a}}{b+2\,c\,x}-\frac {\left (\frac {b\,\left (\frac {b\,\left (\frac {2\,\left (10\,a\,c-b^2\right )}{15\,d^6\,{\left (4\,a\,c-b^2\right )}^3}-\frac {b^2}{10\,d^6\,{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,c}+\frac {4\,b^3-20\,a\,b\,c}{15\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,c}-\frac {4\,a\,b^2-18\,a^2\,c}{15\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\sqrt {c\,x^2+b\,x+a}}{b+2\,c\,x}+\frac {\left (\frac {b\,\left (\frac {b\,\left (\frac {b\,\left (\frac {2\,c\,\left (6\,b^2\,c^2+56\,a\,c^3\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}-\frac {16\,b^2\,c^3}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )}{2\,c}+\frac {2\,c\,\left (11\,b^3\,c-84\,a\,b\,c^2\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )}{2\,c}+\frac {2\,c\,\left (48\,a^2\,c^2+18\,a\,b^2\,c-5\,b^4\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )}{2\,c}+\frac {2\,c\,\left (5\,a\,b^3-24\,a^2\,b\,c\right )}{5\,d^6\,\left (4\,a\,c-b^2\right )\,\left (64\,a\,c^3-16\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^4}-\frac {\left (\frac {b\,\left (\frac {b\,\left (\frac {4\,c^2\,\left (2\,b^2+4\,a\,c\right )}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}-\frac {6\,b^2\,c^2}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}\right )}{2\,c}-\frac {16\,a\,b\,c^2}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}\right )}{2\,c}+\frac {8\,a^2\,c^2}{d^6\,\left (80\,a\,c^3-20\,b^2\,c^2\right )}\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (b+2\,c\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((b*((b*((b*((32*a*c^4)/(3*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2)) - (16*b^2*c^3)/(15*d^6*(4*a*c - b^2)^2
*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8*b*c^2*(6*a*c - b^2))/(3*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2))))/(2
*c) + (4*c*(36*a^2*c^2 - 4*b^4 + 12*a*b^2*c))/(15*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8*a*b
*c*(9*a*c - 2*b^2))/(15*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^2 -
(((b*((b*((2*c^2*(28*a*c - b^2))/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)) - (6*b^2*c^2)/(5*d^6*(4*a*c - b
^2)*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (2*c*(5*b^3 - 28*a*b*c))/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)))
)/(2*c) - (2*c*(5*a*b^2 - 24*a^2*c))/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(
b + 2*c*x)^3 + (((8*a*c - b^2)/(40*c^2*d^6*(4*a*c - b^2)^2) - b^2/(40*c^2*d^6*(4*a*c - b^2)^2))*(a + b*x + c*x
^2)^(1/2))/(b + 2*c*x) - (((b*((b*((2*(10*a*c - b^2))/(15*d^6*(4*a*c - b^2)^3) - b^2/(10*d^6*(4*a*c - b^2)^3))
)/(2*c) + (4*b^3 - 20*a*b*c)/(15*c*d^6*(4*a*c - b^2)^3)))/(2*c) - (4*a*b^2 - 18*a^2*c)/(15*c*d^6*(4*a*c - b^2)
^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b*((2*c*(56*a*c^3 + 6*b^2*c^2))/(5*d^6*(4*a*c - b^2)*(6
4*a*c^3 - 16*b^2*c^2)) - (16*b^2*c^3)/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(11*b^3*c -
84*a*b*c^2))/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(48*a^2*c^2 - 5*b^4 + 18*a*b^2*c))/
(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(5*a*b^3 - 24*a^2*b*c))/(5*d^6*(4*a*c - b^2)*(64*
a*c^3 - 16*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^4 - (((b*((b*((4*c^2*(4*a*c + 2*b^2))/(d^6*(80*a*c^
3 - 20*b^2*c^2)) - (6*b^2*c^2)/(d^6*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (16*a*b*c^2)/(d^6*(80*a*c^3 - 20*b^2*c^
2))))/(2*c) + (8*a^2*c^2)/(d^6*(80*a*c^3 - 20*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^5

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**6,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**
4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b*
*4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(c*x**2
*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 19
2*b*c**5*x**5 + 64*c**6*x**6), x))/d**6

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